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Este trabalho traz resultados de uma investigação que teve como objetivo principal investigar as resoluções de estudantes do 1º ano do ensino médio relativas a atividades de função quadrática, envolvendo a conversão entre os registros de representação simbólico algébrico e gráfico, com o uso do aplicativo GeoGebra em smartphones. Para o desenvolvimento da pesquisa, foi realizada a aplicação de quatro blocos de atividades sobre função quadrática, com foco nas conversões entre os registros algébrico e gráfico, em uma turma de 1º Ano do Ensino Médio de uma escola pública do Paraná. Para a resolução das atividades, os estudantes organizaram-se em dez duplas, um quarteto, e dois estudantes que optaram pela realização individual. Todos os estudantes utilizaram seus próprios smartphones. A organização/seleção das atividades e as análises das produções escritas dos estudantes ocorreram à luz da Teoria dos Registros de Representação Semiótica, de Raymond Duval. Ao final das análises, concluímos que os estudantes indicaram compreender a relação entre cada variável visual do gráfico com a sua correspondente unidade significativa da expressão algébrica, além do avanço nas notações e produções matemáticas escritas, percebido no decorrer da resolução das atividades.

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.

In this study, we aimed to develop a new method for measuring tire-wear particles of less than 2.5 μm generated from vehicle use. We also aimed to devise a method for evaluating the emission factor of tire-wear particles. To develop an evaluation method for tire-wear particles, we examined several factors, such as how tire components in airborne particles collected on a sampling filter were measured, the comparison of tire-wear particles obtained in a laboratory study and an on-road study, a method for measuring tire-wear particles using a test vehicle, and a method for evaluating tire-wear mass using a weighting balance. Measurements of tire-wear particles were carried out using the measurement method proposed herein. The amount of tire wear that the particles generated was almost constant in a vehicle speed range of 20–40 km/h but was influenced by a change in lateral acceleration in the range of 0–0.4G. Furthermore, the relationship between the emission of tire-wear particles and the lateral acceleration force can be shown by a quadratic polynomial. We estimated the emission factor of tire-wear particles by applying the relational equation to the speed profile of the JC08 used in Japanese exhaust gas tests. The emission factor of the test tire used in this study was 3.7 mg/km-vehicle. The ratio of the tire-wear particles to tread wear mass was about 3.3% at PM2.5 and 3.7% at PM10.

Quadratic functions play an important role in applied mathematics. In this paper, we consider the problem of minimizing the integral of a (not necessarily convex) quadratic function in a bounded subset of nonnegative integrable functions defined on a finite-dimensional space that is not compact with respect to any (locally convex) topology in the space of integrable functions. We establish a complete description about the existence or nonexistence of solution in terms of the (strict) copositivity of the matrix involved in the integrand. In addition, we characterize optimality via the Hamiltonian function.

Human eye disease classification adapted by many pieces of research in the last decade due to the importance of eye organ for humans and the evolution in classification techniques. Seven external eye diseases plus normal eye classified in this paper using backpropagation with linear cyclic learning rate-based parabola function. The classification accuracy achieved in this paper is (89.83%).

The extremely useful superelastic behavior of NiTi has been poorly explored because of the limited number of models that can describe the complete hysteretic behavior of NiTi, including a superelastic condition that strongly depends on the applied stress. This paper presents the development of a stress-dependent phenomenological model of NiTi by modifying the existing generalized Prandtl–Ishlinskii (GPI) model. The parameters of the envelop function of the GPI model’s play operator are reformulated as quadratic functions of the applied stress. The stress-dependent GPI model can satisfactorily predict the output strain of a NiTi #6 wire under temperature and stress variation.

In this paper, we consider minimizing a sum of local convex objective functions in a distributed setting, where the cost of communication and/or computation can be expensive. We extend and generalize the analysis for a class of nested gradient-based distributed algorithms (NEAR-DGD; Berahas, Bollapragada, Keskar and Wei, 2018) to account for multiple gradient steps at every iteration. We show the effect of performing multiple gradient steps on the rate of convergence and on the size of the neighborhood of convergence, and prove R-Linear convergence to the exact solution with a fixed number of gradient steps and increasing number of consensus steps. We test the performance of the generalized method on quadratic functions and show the effect of multiple consensus and gradient steps in terms of iterations, number of gradient evaluations, number of communications and cost., Comment: 12 pages, 4 figures. arXiv admin note: text overlap with arXiv:1903.08149

Jigsaw puzzles consist of reconstructing a picture that has been divided into many interlocking pieces. This paper describes an automatic global method for solving the square-piece jigsaw puzzle problem in which neither the orientations nor the locations of the jigsaw pieces are known. This hard combinatorial sorting task is formulated as a nonconvex quadratic programming problem that is solved via the projected power method. Specifically, this work aims to specify the locations and orientations of puzzle pieces by maximizing a constrained quadratic function that resolves an optimized permutation matrix composed of the noisy pairwise affinities between jigsaw pieces. The experimental results obtained in the MIT, McGill and Pomeranz datasets indicate that our method outperforms state-of-the-art techniques.

This paper is concerned with the finite-time annular domain bounded control of Ito-type stochastic systems with Wiener and Poisson random disturbance. First, utilizing different quadratic function methods, some sufficient conditions for finite-time annular domain bounded-ness (FTADB) of the system are achieved. Second, two finite-time annular domain bounded controllers are skillfully developed to ensure the FTADB of the closed-loop system, of which one is state feedback controller and the other is dynamic output feedback controller. Furthermore, an algorithm is provided to deal with the obtained matrix inequalities. Finally, two examples are used to demonstrate the effectiveness of the theorems in this paper.

This paper is concerned with the motor skill learning from human demonstrations using the framework of dynamic regressive models (DRMs). The DRM-based motion planner is preferred as it generates the end-effector trajectory dynamically based on the current state of the end-effector. Within existing frameworks, a single DRM can learn a single motion profile. In addition, such learned DRMs from the data may not be stable. This paper addresses both these issues in a comprehensive manner. In this paper a single DRM has been used to encode human demonstrations involving multitask profiles and multiple task-equilibriums which is novel. We have introduced the idea that the learned DRM will generate human-like stable motion if the energy dissipation rate (EDR) of the generated trajectory follows that of the human demonstration. Thus, the DRM structure has been modified by adding a continuous guiding signal which can be called as the control signal. This signal has been derived using control theoretic principle to ensure asymptotic stability while maintaining the EDR equivalent to that of the human demonstration. The asymptotic stability of the learned DRM has been established by involving a nonmonotonic Lyapunov function consisting of first derivative of a quadratic function and the energy function associated with the DRM. The proposed framework can be learned using many existing regression techniques in this paper Gaussian mixture regression , locally weighted projection regression , and support vector regression techniques have been used successfully. During the pick and place tasks, human demonstrations involving multiple task profiles and multiple task-equilibriums are generated using a 7 DOF commercial robot manipulator. Experimental validations show that the DRMs learned using these three regression schemes are able to guide the robot along the multitask profiles in a stable manner.

We study the global picture of the metric entropy on the space of expanding Blaschke products. We first construct a smooth path in the space tending to a parabolic Blaschke product. We prove that the metric entropy on this path tends to 0 as the path tends to this parabolic Blaschke product. It turns out that the limiting parabolic Blaschke product on the unit circle is conjugate to the famous Boole map on the real line. Thus we can give a new explanation of Boole's formula discovered more than one hundred and fifty years ago. We modify the first smooth path to get a second smooth path in the space of expanding Blaschke products. The second smooth path tends to a totally degenerate map. We see that the first and second smooth paths have completely different asymptotic behaviors near the boundary of the space of expanding Blaschke products. However, they represent the same smooth path in the space of all smooth conjugacy classes of expanding Blaschke products. We use this to give a complete description of the global graph of the metric entropy on the space of expanding Blaschke products. We prove that the global graph looks like a bell. It is the first result to show a global picture of the metric entropy on a space of hyperbolic dynamical systems. We apply our results to the measure-theoretic entropy of a quadratic polynomial with respect to its Gibbs measure on its Julia set. We prove that the measure-theoretic entropy on the main cardioid of the Mandelbrot set is a real analytic function and asymptotically zero near the boundary.

Interaction solutions between lump and soliton are analytical exact solutions to nonlinear partial differential equations. The explicit expressions of the interaction solutions are of value for analysis of the interacting mechanism. We analyze the one-lump-multi-stripe and one-lump-multi-soliton solutions to nonlinear partial differential equations via Hirota bilinear forms. The one-lump-multi-stripe solutions are generated from the combined solution of quadratic functions and N exponential functions, while the one-lump-multi-soliton solutions from the combined solution of quadratic functions and N hyperbolic cosine functions. Within the context of the derivation of the lump solution and soliton solution, necessary and sufficient conditions are presented for the two types of interaction solutions, respectively, based on the combined solutions to the associated bilinear equations. Applications are made for a (2+1)-dimensional generalized KdV equation, the (2+1)-dimensional NNV system and the (2+1)-dimensional Ito equation.

In this study, results of parametric effects and optimization of turbidity removal from produced water using response surface methodology (RSM) and artificial neural network (ANN) based on a statistically designed experimentation via the Box–Behnken design (BBD) are reported. A three-level, three-factor BBD was employed using dosage ( x 1 ), time ( x 2 ) and temperature ( x 3 ) as process variables. A quadratic polynomial model was obtained to predict turbidity removal efficiency. The RSM model predicted an optimal turbidity removal efficiency of 83% at conditions of x 1 (1 g/L), x 2 (16.5 min) and x 3 (45 °C) and validated experimentally as 82.73% with low model lack of fit F value of 0.6 and CV value of 8.22%. The ANN model predicted optimal turbidity removal of 83.01% at conditions of x 1 (1 g/L), x 2 (16.5 min) and x 3 (45 °C) and validated as 82.98%. Both models showed to be effective in describing the parametric effect of the considered operating variables on the turbidity removal from produced water. However, the ANN described the parametric effect more accurately when compared with the RSM model, with a smaller PRE (percentage relative error) and AAD (absolute average deviation) of ±0.0241% and ±0.0139%, respectively.

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.

This paper is concerned with the stability analysis problems of linear systems with time-varying delays using integral inequalities. To reduce the conservatism of stability criteria obtained with Lyapunov-Kraosvksii approach, there has been a growing tendency to utilize various integral quadratic functions in the construction of Lyapunov-Krasovskii functionals. Consequently, integral inequalities also have played key roles to derive stability criteria guaranteeing the negativity of the Lyapunov-Krasovskii functional’s derivative. Recently, by utilizing first-degree or second-degree orthogonal polynomials, new free-matrix-based integral inequalities have been proposed for integral quadratic functions containing both a system state variable and its time derivative. This paper tries to generalize these inequalities and their stability criteria with a note on the relation among the existing integral inequalities and the proposed one. In this note, it is shown that increasing a degree of the proposed integral inequality only reduces or maintains the conservatism by deriving the hierarchical stability criteria. Four numerical examples including practical systems demonstrate the effectiveness of the proposed methods in terms of allowable upper delay bounds.

The authors introduce a finite variable quadratic functional equation and derive its solution. Also, authors examine its Hyers-Ulam stability in Random Normed space(RN-space) by means of two different methods.

In this article, the repetitive finite-length linear discrete-time singular system is formulated as an input-output equation by virtue of the lifted-vector method and a gain-optimized P-type iterative learning control profile is architected by sequentially arguing the learning-gain vector in minimizing the addition of the quadratic norm of the tracking-error vector and the weighed quadratic norm of the compensation vector. By virtue of the elementary permutation matrix and the property of the quadratic function, the optimized-gain vector is solved and explicitly expressed by the system Markov matrix and the iteration-wise tracking error. Then the linearly monotonic convergence of the tracking error is derived under the assumption that the initial state of the dynamic subsystem is resettable. Furthermore, for the circumstance that the system parameters uncertainties exist, the quasi scheme is established by replacing the exact system Markov matrix with the approximated one in the optimized gain. Rigorous analysis conveys that the proposed gain-optimized scheme is robust to the system internal disturbance within a suitable range. The validity and effectiveness are demonstrated numerically.

In this paper, a new method to find quadratic function solutions to bilinear forms is proposed. By applying the Hirota direct method, we construct some important exact solutions to the B-type Kadomtsev-Petviashvili (BKP) equation of fourth-order. Solitons, rational solutions, lump solutions and interaction solutions are presented with the help of symbolic computations. The dynamics of some selected solutions are also studied with the aid of 3D plots.

This paper deals with a robust stability problem for uncertain Lur’e systems with time-varying delays and sector-bounded nonlinearities. An improved delay-dependent robust stability criterion is proposed via a modified Lyapunov-Krasovskii functional (LKF) approach. Firstly, a modified LKF consisting of delay-dependent matrices and double-integral items under two delay subintervals is constructed, thereby making full use of the delay and its derivative information. Secondly, the stability criteria can be expressed as convex linear matrix inequality (LMI) via the properties of quadratic function application. Thirdly, to further reduce the conservatism of stability criteria, the quadratic generalized free-weighting matrix inequality (QGFMI) is used. Finally, some numerical examples, including the Lur’e system and the general linear time-delayed system, are presented to show the improvement of the proposed approach.

We theoretically analyze the information geometric optimization (IGO), which is a unified framework of stochastic search algorithms for black-box optimization. The IGO framework has two parameters: 1) the learning rate and 2) the sample size, and they influence the behavior of the algorithm. We investigate the strategy parameters of the IGO with the family of isotropic Gaussian distributions on a general convex quadratic function. Compared to the previous theoretical works, where an infinite sample size is assumed and the deterministic algorithm dynamics is studied, we investigate the expected improvement of the algorithm with a finite sample size. The analysis finds that the relative decrease rates of the distance from the distribution mean to the landscape optimum and the distribution standard deviation must be the same, which we observe in practice, while the analysis based on an infinite sample size failed to obtain. We derive these rates explicitly as a function of the eigenvalues of the Hessian of the objective function and the strategy parameters. We also derive the stable value of the ratio of the square distance to the optimum over the distribution variance, as well as the conditions that the stable value exists. These theoretical values coincide with our numerical simulations.

This study presents a framework for the enhancement of parameter estimation issues by computing the time average of several quantum observables measured under the evolution of the Hamiltonian system. These time-cum-quantum averages are expressed as a quadratic function of the orthogonal projections on the eigen subspaces of the system. Estimation of the parameters is done using the approximate time-independent perturbation theory assuming the other parameters to be small with respect to one of the parameters that is normalised to unity in the presence of noise. The study also proves that the derived expression for the mean square parameter estimation error attains the quantum limit. When the measurement noise is classical and for general positive operator-valued measure successive time, the authors derive an expression for the quantum Cramer–Rao lower bound.

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.

A formulation, based on indirect approach, that uses both the aerodynamic angles: angle of attack and bank angle as control variables and maximizes the parachute deployment altitude of a Mars entry vehicle is presented. The complexity of handling the control variable ‘angle of attack’ in the indirect approach is overcome by expressing the aerodynamics coefficients as a quadratic polynomial of angle of attack. The problem is formulated as a two point boundary value problem using the Pontryagin’s principle of the optimal control theory. The solution is obtained using differential evolution technique, a heuristic optimization technique. This is an alternative formulation to the commonly used direct approach using non-linear programming. The solution procedure based on indirect approach reduces the number of unknowns drastically compared to the direct approach. The benefit of using angle of attack modulation in addition to bank angle modulation is quantified. The implication of constraints on minimum allowable altitude and maximum deceleration on the optimized trajectory is analyzed using the new formulation and the solution approach.

In this paper, we explore the qualitative recognition by 144 high school students of the visual variables of the graphical representation of a quadratic function and the significant symbolic units of its algebraic writing. This recognition emerges when students are asked to move from the graphical register to the algebraic register, and conversely. The results show that the association between the visual variables of the graphic register and the symbolic units of algebraic writing has not been fully recognized. It is considered that, although the students study the effects of the coefficients of a quadratic function, they tend to only associate them with translations through the axes, instead of with pertinent visual changes. This is not sufficient for a qualitative global apprehension that allows the coordination between the visual variables of the graphical representation of a quadratic function and the significant symbolic units of its algebraic representation. Dans cet article on explore la reconnaissance qualitative, par 144 élѐves de baccalauréat, des variables visuelles de la représentation graphique d’une fonction quadratique et des unités symboliques de sa représentation algébrique. Cette reconnaissance se manifeste quand on demande aux élѐves de passer du registre graphique à l’algébrique et vice-versa. Les résultats montrent que l'association entre des variables visuelles du registre graphique et des unités symboliques de l'écriture algébrique n'a pas été complètement reconnue. Bien que les élèves aient abordé l’étude des effets des coefficients d’une fonction quadratique, on observe qu’ils tendent à ne leur associer que des translations selon les axes, au lieu des changements visuels pertinents. Cela n'est pas suffisant pour une appréhension globale qualitative, qui permet la coordination entre les variables visuelles de la représentation graphique et les unités symboliques de la représentation algébrique d’une fonction quadratique. En este artículo se explora el reconocimiento cualitativo que hacen 144 estudiantes de bachillerato sobre las variables visuales de la representación gráfica y las unidades simbólicas de la representación algebraica de una función cuadrática. Este reconocimiento se manifiesta cuando se pide a los estudiantes pasar del registro gráfico al algebraico, y viceversa. Los resultados muestran que la asociación entre las variables visuales del registro gráfico y las unidades simbólicas de la escritura algebraica no ha sido completamente reconocida. Se considera que, si bien los estudiantes abordan el estudio del comportamiento de los coeficientes de la función cuadrática, tienden a sólo asociarlos con traslaciones a través de los ejes, en lugar de variables visuales pertinentes. Lo cual no es suficiente para una aprehensión global cualitativa que permite la coordinación entre las variables visuales de la representación gráfica y las unidades simbólicas de la representación algebraica de la función cuadrática.

In this paper, we study a Hamilton–Jacobi flow { H t τ } t > 0 starting from the Takagi function τ . The Takagi function is well known as a pathological function that is everywhere continuous and nowhere differentiable on R . As the first result of this paper, we derive an explicit representation of { H t τ } . It turns out that H t τ is a piecewise quadratic function at any time and that the points of intersection between the parabolas are given in terms of binary expansion of real numbers. Applying the representation formula, we next give the main result, which asserts that { H t τ } has a self-affine property of evolutional type involving a time difference in the functional equality. Furthermore, we determine the optimal time until when the self-affine property is valid.

This paper focuses on a class of zero-norm composite optimization problems. For this class of nonconvex nonsmooth problems, we establish the Kurdyka–Łojasiewicz property of exponent being a half for its objective function under a suitable assumption and provide some examples to illustrate that such an assumption is not very restricted which, in particular, involve the zero-norm regularized or constrained piecewise linear–quadratic function, the zero-norm regularized or constrained logistic regression function, the zero-norm regularized or constrained quadratic function over a sphere.

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.

The generation unit aggregation is frequently used in a virtual power plant which participates in real-time spot markets. In this letter, an analytical method is proposed to quickly and precisely compute a piecewise quadratic function of aggregated units. Three propositions are introduced and proven in the letter in order to guarantee the accuracy of the equivalent model after aggregation. Numerical results verify the validity of the proposed method.

This article presents the exact evaluation of the physical optics (PO) integral in time-domain. The surface of a perfect electrically conducting (PEC) scatterer is modeled by quadratic triangles. Using the Radon transform interpretation, the PO integral is reduced to a line integral, which is formed by the intersection of the quadratic surface and the plane defined by the Dirac delta function. It is shown that the resulting line is a quadratic curve, e.g., an ellipse, in barycentric coordinates of the quadratic triangle, since the incident wave is a plane wave and the scattered fields are observed at the far-field. To evaluate the PO integral: 1) the PO integral is represented in barycentric coordinates and the type of the intersecting curve is determined; 2) an appropriate coordinate transformation is applied, e.g., elliptic coordinates for the ellipse; and then 3) the Gauss–Legendre quadrature rule (GLQR) is applied. It is shown that a suitable GLQR order yields the PO integral exactly. The validity and accuracy of the proposed method are demonstrated via numerical examples.

In recent work, we use Dudek’s method together with a result of Zagier to establish an asymptotic formula for the average number of divisors of an irreducible quadratic polynomial of the form $$x^{2}-bx+c$$ with b, c integers. In this note, we remark that one can adopt the work of Hooley to derive a more precise asymptotic formula for the case $$x^{2}-bx+c$$ with $$b^{2}-4c$$ not a square, and as a consequence, re-establish the weaker asymptotic formula given in our recent work by different arguments.

This article studies the stability problems of linear systems with time-varying delay. An improved negativity condition for a quadratic function parameterised by time-varying delay is derived. Based on this condition, less conservative stability criteria are obtained. Through two well-known numerical examples, it is proved that the proposed method can reduce the conservativeness of the computed results.

In this study, the authors propose for the parameter estimation of chirp signals at low signal-to-noise ratios using the spectrum phase instead of the traditional temporal phase. First, the spectrum phase expression of chirp signals is derived, which demonstrates that the spectrum phase can be approximated as a quadratic function of the frequency. However, the spectrum phase is a wrapped phase modulo 2π, like the temporal phase. To obtain the unwrapped spectrum phase, the authors develop a robust phase unwrapping method and then perform linear regression on the unwrapped spectrum phase to obtain the estimates of the parameters. Finally, the estimates are refined by the O'Shea refinement strategy to reduce bias. The simulation results show that the proposed technique has a lower signal-to-noise ratio threshold and is less complex than the existing methods.

In this paper, we propose non-model-based strategies for locally stable convergence to Nash equilibrium in quadratic noncooperative games where acquisition of information (of two different types) incurs delays. Two sets of results are introduced: (a) one, which we call cooperative scenario, where each player employs the knowledge of the functional form of his payoff and knowledge of other players’ actions, but with delays; and (b) the second one, which we term the noncooperative scenario, where the players have access only to their own payoff values, again with delay. Both approaches are based on the extremum seeking perspective, which has previously been reported for real-time optimization problems by exploring sinusoidal excitation signals to estimate the Gradient (first derivative) and Hessian (second derivative) of unknown quadratic functions. In order to compensate distinct delays in the inputs of the players, we have employed predictor feedback. We apply a small-gain analysis as well as averaging theory in infinite dimensions, due to the infinite-dimensional state of the time delays, in order to obtain local convergence results for the unknown quadratic payoffs to a small neighborhood of the Nash equilibrium. We quantify the size of these residual sets and corroborate the theoretical results numerically on an example of a two-player game with delays.

We investigate Diophantine problems concerning linear combinations of polynomials of the shape $$a_0 x+a_1x(x+1)+ a_2x(x+1)(x+2)+ \cdots + a_n x(x+1)\ldots (x+n)$$ with $$n\in \mathbb{N}\cup\{0\}$$ . We provide effective finiteness results for the power, shifted power, and quadratic polynomial values of these linear combinations, generalizing the analogous results of Hajdu, Laishram and Tengely [10], and of Berczes, Hajdu, Luca and the author [2] given for the sums $$x+x(x+1)+x(x+1)(x+2)+ \cdots + x(x+1)\ldots (x+n)$$ , i.e., for the case $$a_0=a_1= \cdots = a_n = 1$$ . Our work is closely connected also with some results of Tengely and Ulas [15] concerning the case when the coefficients $$a_0,a_1, \ldots, a_n$$ are zeroes and ones.

This paper focuses on designing a unified approach for computing the projection onto the intersection of a one-norm ball or sphere and a two-norm ball or sphere. We show that the solutions of these problems can all be determined by the root of the same piecewise quadratic function. We make use of the special structure of the auxiliary function and propose a novel bisection algorithm with finite termination. We show that the proposed method possesses quadratic time worst-case complexity. The efficiency of the proposed algorithm is demonstrated in numerical experiments, which show the proposed method has linear time complexity in practice.

The celebrated Motzkin–Straus formulation for the maximum clique problem provides a nontrivial characterization of the clique number of a graph in terms of the maximum value of a nonconvex quadratic function over a standard simplex. It was originally developed as a way of proving Turan’s theorem in graph theory, but was later used to develop competitive algorithms for the maximum clique problem based on continuous optimization. Clique relaxations, such as s-defective clique and s-plex, emerged as attractive, more practical alternatives to cliques in network-based cluster detection models arising in numerous applications. This paper establishes continuous cubic formulations for the maximum s-defective clique problem and the maximum s-plex problem by generalizing the Motzkin–Straus formulation to the corresponding clique relaxations. The formulations are used to extend Turan’s theorem and other known lower bounds on the clique number to the considered clique relaxations. Results of preliminary numerical experiments with the CONOPT solver demonstrate that the proposed formulations can be used to quickly compute high-quality solutions for the maximum s-defective clique problem and the maximum s-plex problem. The proposed formulations can also be used to generate interesting test instances for global optimization solvers.

Here, we introduce the quadratic orthogonal projection (the problem of projecting a point onto a quadratic level set), as an extension of the linear orthogonal projection (the problem of projecting...