Quadratic Function Research Articles

Stability analysis of discrete-time delayed systems via matrix-injection-based transformation method for bivariate quadratic functions

In this paper, the stability analysis of discrete-time systems with a time-varying delay is studied. First, a selectable matrix-injection-based transformation lemma is proposed to guarantee the negative (or positive) definiteness of bivariate quadratic function. This lemma allows a selection of conditions that combine different delay sets, which is more relaxed than that in most previous studies. Then, an improved Lyapunov–Krasovskii functional (LKF) with delay-squared-product summation is given, which introduces extra cross-terms about delay-variation square for stability analysis. Considering the role of injection matrices in the proposed lemma, together with two types of allowable delay sets, four stability criteria are established. Finally, the less conservatism of the proposed criteria is verified by two numerical examples, which demonstrate the superiority of the presented methods.

Discrete‐time linear quadratic gaussian control with input delay and Markovian packet dropouts

AbstractThis article investigates the finite horizon linear quadratic gaussian (LQG) control problem for networked control systems with two‐way Markovian packet dropouts and input delay, where packet dropouts occur from the sensor to the estimator and from the controller to the actuator, and delay occurs from the controller to the actuator. The novelty of this work lies in providing a complete solution for the optimal control problem subject to two‐way Markovian packet dropouts and input delay. The contributions of this article are as follows: first, by applying the maximum principle to the discrete‐time linear systems and the quadratic cost function involving input delay and Markovian packet dropouts, a solution to the forward and backward stochastic difference equations (FBSDEs) is derived. Second, a sufficient and necessary condition for the optimal control problem is obtained by decoupling the coupled Riccati equations. Finally, the explicit solution of the optimal controller is presented based on the complete square method. Numerical examples are provided to demonstrate the validity of the theoretical results.

A Dual Bounding Framework Through Cost Splitting for Binary Quadratic Optimization

Binary quadratic programming (BQP) is a class of combinatorial optimization problems comprising binary variables, quadratic objective functions, and linear/nonlinear constraints. This paper examines a unified framework to reformulate a BQP problem with linear constraints to a new BQP with an exponential number of variables defined on a graph. This framework relies on the concept of stars in the graph to split the quadratic costs into adjacent and nonadjacent components indicating in-star and out-of-star interactions. We exploit the star-based structure of the new reformulation to develop a decomposition-based column generation algorithm. In our computational experiments, we evaluate the performance of our methodology on different applications with different quadratic structures. The quadratic component of the problem is dealt with in the column generation master problem and its subproblem. Results indicate the superiority of the framework over one of the state-of-the-art solvers, GUROBI, when applied to various benchmark reformulations with adjacent-only or sparse quadratic cost matrices. The framework outperforms GUROBI in terms of both dual bound and computational time in almost all instances. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete. Funding: The authors thank the Mitacs Accelerate Program for providing funding for this project. In addition, B. Rostami gratefully acknowledges the funding provided by the Canadian Natural Sciences and Engineering Research Council (NSERC) under a Discovery Grant [Grant RGPIN-2020-05395]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2021.0186 .

Soil organic carbon enrichment in the particulate matter emitted by rural soils: A laboratory assessment

AbstractThe aim of this study was to assess the organic carbon (OC) content in the PM10 emitted by agricultural soils and rural roads under controlled conditions. Samples were collected from agricultural soils and rural roads. The PM10 was generated and collected using an electrostatic precipitator coupled with the Easy Dust Generator (EDG). This procedure ensures that the PM10 collected come specifically from soil. OC contents were measured in both the soil and PM10. The enrichment ratio (ER) was calculated as the ratio between OC content in the PM10 and OC content in the soil. The results showed that OC content in the PM10 ranged from 2.7% to 3.5% in agricultural soils and from 1.4% to 2.9% in rural roads. These values were comparable with the OC contents observed in fine particles transported by the wind, but lower than OC contents observed in PM10 samples collected in rural areas using active samplers and filters. A quadratic function described the association between OC in PM10 and OC in the soil. A negative potential function described the association between ERs and OC in the soil. Both associations suggested a saturation of OC in PM10 when the OC content in the soil was high. This information is crucial for a better comprehending of the dust emission role in the redistribution of OC within terrestrial ecosystems and to the atmosphere and oceans.

Stabilization of continuous-time Markovian jump systems: A mode separation but optimization method

This paper addresses the stabilization problem of continuous-time Markovian jump systems (MJSs) by applying an optimization controller. A new stabilizing method based on a mode separation algorithm is proposed, whose separation will be optimized by minimizing the established cost function value. It will be seen that the presented optimal controller will have hybrid decision variables such as continuous and discrete variables. Meanwhile, the related cost function is in contrast to the traditionally linear quadratic functions. The conditions for the existence of the developed controller are established via using rigorous theory analysis, and its detailed computation method is presented too. It can be concluded that traditionally mode-dependent and -independent controllers about MJSs are contained as two special situations. A practical example is offered so as to verify the effectiveness and superiority of the methods proposed in this study.

Thermal contraction coordination behavior between unbound aggregate layer and asphalt mixture overlay based on the finite difference and discrete element coupling method

AbstractThe constraint action of the unbound aggregate layer underneath plays an important role in affecting the temperature strains in the top asphalt layer. The focus of the present paper is to investigate the interactive thermal contraction mechanisms between the asphalt mixture and granular base layers to offer a new perspective in promoting the understanding of the thermal cracking disease. In this paper, both experiments and simulations were conducted for the thermal contraction tests. A novel numerical modeling of virtual composite structures was proposed using wall‐zone coupling operation based on the finite difference and discrete element coupling method. The experimental composite structures containing three types of asphalt mixture overlays and five types of unbound aggregate base layers were developed for verification. The thermal contraction and restraint mechanism were revealed from both macro and microscopic scales. The proposed numerical modeling shows a 94% high accuracy. The particle displacement vector and contact force chain could explain the thermal contraction behavior of composite structures. Smaller particle motion displacement or stronger contact force chain result in a higher restraint strain of the asphalt overlay. The thermal contraction behavior can be coordinated through the compaction and loosening of unbound aggregates. The material parameters and cooling temperature differences in the asphalt overlay have slight effects on the constraint action of a base layer, while the gradation and mechanical parameters of the unbound aggregate layer show significant impaction. The parameters, cohesion C and friction angle φ, show a quadratic function with the restraint coefficient. This work has significant guidance on the selection of pavement structure and materials to improve the thermal cracking problem and lay the basis for the mechanical theoretical calculation to predict thermal cracking.

Higher-order galerkin finite element method for nonlinear coupled reaction-diffusion models

In this article, generalized nonlinear coupled reaction-diffusion (NCRD) models are analyzed using the higher-order Galerkin finite element method (FEM). The authors present finite element analysis for one- and two-dimensional NCRD models by applying quadratic Lagrange basis functions. Optimal order convergence in space is proved by applying quadratic FEM in spatial discretization and utilizing the Crank-Nicolson (CN) scheme for time discretization. To address the nonlinearity of NCRD models, CN extrapolation and predictorcorrector schemes are employed. The stability analysis for generalized NCRD models and the applied CN scheme are discussed by utilizing the eigenvalues method. Finally, to assess the accuracy and efficiency of the proposed scheme and to capture different patterns of NCRD models, various one- and two-dimensional models are considered. The obtained results are numerically and graphically compared with existing literature. It is observed that quadratic basis functions in FEM improve accuracy and efficiency of the results.

Relationship among runoff, soil erosion, and rill morphology on slopes of overburdened stockpiles under simulated rainfall

Overburdened stockpiles are difficult to achieve natural stability in a short period time because of their complex and diverse composition of materials and significant differences in physical and chemical properties, causing numerous cases of water loss and soil loss events. We conducted a series of experiments at different rainfall intensities (30, 60, 90, and 120 mm h−1) and different gravel contents (0 %, 10 %, 20 %, 30 %, and 40 %) to study the runoff, soil erosion, and rill morphology characteristics and relationship among three of overburdened stockpiles in production and construction projects with simulated rainfall. The runoff rate and soil loss rate basically showed a trend of increasing and then decreasing with gravel content and lower gravel content had a certain degree of inhibition on the flow velocity. 20 %, 0 %, 0 %, and 20 % gravel contents had the most significant inhibition on the flow velocity at 30, 60, 90, and 120 mm h−1 rainfall intensities, respectively. Flow shear stress, stream power, unit energy of water cross-section, maximum rill length, width and depth all increased with increasing rainfall intensity. The stream power, unit stream power, and unit energy of water cross-section were larger at higher gravel content. The maximum rill length and depth were positively correlated with gravel content, which were the minimum at 0 % gravel content and all rainfall intensities. Gravel content played a more important role in the runoff, soil loss, and rill morphology. The relationship between runoff rate and soil loss rate, flow velocity and rill morphology parameters were all expressed as linear functions, and flow velocity and soil loss rate was quadratic function, with a correlation coefficient of 0.339. In contrast, runoff rate and rill morphology parameters, soil loss rate and rill morphology parameters, erosion dynamics parameters and rill morphology parameters were all expressed as power functions. In addition, the runoff rate can better describe the process of soil loss (R2 = 0.798, P < 0.05). The unit energy of water cross-section was more suitable to describe the development of maximum rill length and depth (with rill length was R2 = 0.781, and with rill depth was R2 = 0.723, P < 0.05). At the same time, the stream power was more suitable to describe the development of maximum rill width on the slope of overburdened stockpiles (R2 = 0.841, P < 0.05). These results contributed to preventing soil loss on overburdened stockpiles and helping the development of estimation models for soil erosion.

Quadratic Growth of Out-of-Time-Ordered Correlators in Quantum Kicked Rotor Model.

We investigate both theoretically and numerically the dynamics of out-of-time-ordered correlators (OTOCs) in quantum resonance conditions for a kicked rotor model. We employ various operators to construct OTOCs in order to thoroughly quantify their commutation relation at different times, therefore unveiling the process of quantum scrambling. With the help of quantum resonance condition, we have deduced the exact expressions of quantum states during both forward evolution and time reversal, which enables us to establish the laws governing OTOCs' time dependence. We find interestingly that the OTOCs of different types increase in a quadratic function of time, breaking the freezing of quantum scrambling induced by the dynamical localization under non-resonance condition. The underlying mechanism is discovered, and the possible applications in quantum entanglement are discussed.

Integrability, breather, rogue wave, lump, lump-multi-stripe, and lump-multi-soliton solutions of a (3 + 1)-dimensional nonlinear evolution equation

In this article, we consider a new (3 + 1)-dimensional evolution equation, which can be used to interpret the propagation of nonlinear waves in the oceans and seas. We effectively investigate the integrable properties of the considered nonlinear evolution equation through several aspects. First of all, we present some elementary properties of multi-dimensional Bell polynomial theory and its relation with Hirota bilinear form. Utilizing those relations, we derive a Hirota bilinear form and a bilinear Bäcklund transformation. By employing the Cole–Hopf transformation in the bilinear Bäcklund transformation, we present a Lax pair. Additionally, using the Bell polynomial theory, we compute an infinite number of conservation laws. Moreover, we obtain one-, two-, and three-soliton solutions explicitly from Hirota bilinear form and illustrate them graphically. Breather solutions are also derived by employing appropriate complex conjugate parameters in the two-soliton solution. Choosing the generalized algorithm for rogue waves derived from the N-soliton solution, we directly obtain a first-order center-controllable rogue wave. Lump solutions are formulated by employing a well-established quadratic test function as a solution to the Hirota bilinear form. Further taking the test function in a combined form of quadratic and exponential functions, we obtain lump-multi-stripe solutions. Furthermore, a combined form of quadratic and hyperbolic cosine functions produces lump-multi-soliton solutions. The fission and fusion effects in the evolution of lump-multi-stripe solutions and lump-soliton-solutions are demonstrated pictorially.

Estimation of the residue capacity of lithium iron phosphate battery based on the internal resistance obtained from charging voltage drop

This study takes the 80 Ah lithium iron phosphate (LFP) prismatic battery that is from the vehicle and is in the middle or end of life as the research target, and the voltage-drop resistance (VDR), which is calculated through the voltage drop at the charging end, is used to for residue capacity estimation. The relationship between the VDR and the residue capacity of the battery is systematically studied, including factors such as charging cut-off voltage, charging current, charging end time, and charging current change mechanism. The results show that the residue capacity is a quadratic function relationship with the charging cut-off voltage and VDR. The VDR is affected by the charging cut-off voltage, charging cut-off current, and charging end time. There is no significant difference in the influence of VDR obtained by the charging modes of step charging and constant current charging. By testing the VDR under different charging conditions and different resting times, and standardizing the VDR, the residue capacity estimation error of 97% of the vehicle’s batteries is within ±5%, which meets the application requirements for residue capacity estimation.

Analysis of the Fracture Characteristics and Crack Propagation Mechanism of Fractured Sandstone under Dynamic Loading

In order to study the impact of fracture thickness on the dynamic mechanical properties of rock specimens, impact compression tests on prefabricated sandstone specimens with different fracture thicknesses and intact specimens were carried out by using a split Hopkinson pressure bar (SHPB) test device, and the dynamic mechanical parameters of the fractured sandstone specimens were obtained. The results showed that the dynamic stress–strain curves of the prefabricated fractured sandstone specimens are similar to those of the intact sandstone specimens, which can be divided into three stages: elasticity, plasticity, and failure. With the increase in the thickness of prefabricated cracks, the dynamic compressive strength of the sandstone specimens decreases in a quadratic function, the dynamic strain decreases in a power exponential function, and the dynamic elastic modulus decreases linearly. Attempts were made to quantitatively analyze the crushing degree of sandstone specimens. The average particle size of the crushed specimens was negatively correlated with the thickness of prefabricated cracks, and the fractal dimension was linearly negatively correlated with the dynamic compressive strength. The 1.5–2.5 mm prefabricated fractured sandstone specimens produced airfoil cracks and secondary cracks; the 3–3.5 mm prefabricated fractured specimens produced airfoil cracks, coplanar secondary cracks, and secondary oblique cracks; and the complete specimens were subjected to axial failure to produce axial cracks.

Strain-rate dependent mixed-mode traction laws for glass fiber-epoxy interphase using molecular dynamics simulations

In this paper, we establish a methodology to predict strain rate-dependent mixed-mode traction-separation responses (i.e., traction laws) for glass-epoxy composite interphase using molecular dynamics (MD) simulations. Glass-epoxy interphases with monolayer glycidoxypropyltrimethoxy silane are prepared by varying silane number density from 0.0 nm−2 to 3.9 nm−2 following the epoxy-amine diffusion and curing reactions. To established the effects of strain rate and mode-mixity on the interphase traction laws, the nano-meter size interphase domain is loaded in various mode-mixity (θ=0°(Mode−II),15°,30°,45°,60°,and90°(Mode−I)) with full range of strain rates from quasit-static to high strain rate (∼1e16/s) where a theoretical plateau strength limit is predicted. Following our previous work on Mode-I [Chowdhury et al., Composites Part B 237 (2022) 109877], mathematical model is developed for Mode-II as function of strain rate for different interphase structures (i.e., silane number density). The continuum equivalent bi-linear cohesive traction law is developed using the MD results to determine the mode-mixity quadratic functions and associated exponents for peak tractions, energy absorption, crack initiation and crack opening displacement from the mixed-mode simulations data. The MD predicted traction laws can be used to model interphase in micromechanics finite element analysis to bridge the length scale for the prediction of fiber-matrix debonding in composites.

A unified empirical method for predicting both vertical and horizontal ground displacements induced by tunnel excavation

An empirical model for describing the soil movements induced by tunneling is proposed, then the mathematical relationship between horizontal displacement and vertical displacement is obtained. By analyzing 25 sets of data from field engineering and 35 sets of data from model tests, the formula for the maximum settlement Sv, max( z) is optimized to adapt to different ground conditions. The Modified Gaussian function is developed by introducing the existing width coefficient of the settlement trough i( z) and the optimized Sv, max( z) to describe the surface and subsurface vertical displacement. Subsequently, the formula for the horizontal displacement is derived based on the Modified Gaussian function. H( z) representing the position of the oriented point of the soil movement is a variable in the formula for horizontal displacement. Based on measured results, a logarithmic function and a quadratic polynomial function are proposed to describe the variation of H( z) with depth in the clay stratum and sand stratum, respectively. Then, the rationality of the proposed method is validated by 4 sets of in-situ data and 4 sets of test data. Finally, the implementation process of the proposed method is illustrated, and the inversion results of the ground displacement field in Beijing Metro Line 12 are presented.

Estimating Yield Response Functions to Nitrogen for Annual Crops in Iran

Nitrate is a crucial element for crop growth, and its optimal application is essential for maximizing agricultural yield. In Iranian agriculture, there is a substantial gap between recommended nitrate usage and what farmers actually apply. In this study, our primary objective is to determine the most effective utilization of nitrate for crop cultivation. Simultaneously, we aim to analyze the factors that contribute to the disparity between optimal and current nitrate application practices. Furthermore, our research explores the impact of these differences on regional variations in crop yields. This is achieved using a quadratic yield response function model based on unbalanced panel data spanning the years 2000 to 2016, which includes a total of 14 crop activities and encompasses 31 administrative regions. The results show that rice exhibits the highest nitrogen usage, while rain-fed wheat demonstrates the lowest utilization at the optimal point. Depending on whether random- or fixed-effects estimation is found to be the most suitable specification, average yields corresponding to the optimal level of nitrogen use are calculated by region, or the average across all regions. In Iran, the top-performing regions for cereals like rain-fed wheat and irrigated barley can achieve yields of 1.33 and 3 t/ha, respectively. These yields represent a 31% and a 9% increase from the levels observed in 2016. The outcomes derived from the estimated yield response function will be integrated into comprehensive agricultural, economic, and environmental optimization models. These integrated models will facilitate the assessment of various fertilizer policies on fertilizer use, land allocation, farm-household incomes, and environmental externalities, such as nitrate leaching and nitrate balance. This study holds substantial scientific promise, given its exploration of the policy implications surrounding fertilizer usage, making it crucial not only for Iran, but also for many developing nations grappling with inefficient and unsustainable agricultural practices. It represents the first of its kind in the literature, providing estimations of optimal nitrogen use and crop yield points across all regions in Iran. This is achieved through advanced visualization using GIS maps.

Experimental Study on Dynamic and Static Combined Dynamics of Temperature–Water-Coupled Sandstone and Energy Consumption Analysis

In order to study the dynamic properties of temperature–water-coupled sandstone under axial pressure, impact compression tests were carried out on sandstone samples after temperature–water coupling under eight types of axial pressure (0.5~4.0 MPa) loading as well as no axial pressure loading by using the split-Hopkinson pressure bar (SHPB) test set. The results showed that the mass, volume, and density of the sandstone specimens increased by 0.57%, 0.37%, and 0.20%, respectively, after temperature–water coupling. With increasing axial pressure, the dynamic compressive strength of temperature–water-coupled sandstone samples decreased as a linear function, the dynamic strain increased as a quadratic function, the dynamic modulus of elasticity decreased as a quadratic function, and the average strain rate increased as an exponential function, indicating a strong strain rate effect. From the energy point of view, as the axial pressure increases, the absorption energy of the sample increases, the reflection energy gradually decreases, the crushing degree of the sample increases, and the size of the broken pieces decreases; the average particle size of the sandstone sample pieces decreases quadratically with the increase in the absorption energy and linearly with the increase in the axial pressure.

Optimal investment problem for a hybrid pension with intergenerational risk-sharing and longevity trend under model uncertainty

This paper studies the optimal investment problem for a hybrid pension plan under model uncertainty, where both the contribution and the benefit are adjusted depending on the performance of the plan. Furthermore, an age and time-dependent force of mortality and a linear maximum age are considered to capture the longevity trend. Suppose that the plan manager is ambiguity averse to the financial market and is allowed to invest in a risk-free asset and a risky asset. The plan manager aims to find optimal investment strategies and optimal intergenerational risk-sharing arrangements by minimizing the unstable contribution risk, the unstable benefit risk and the discontinuity risk under the worst-case scenario. By applying the stochastic optimal control approach, robust optimal strategies are derived under the worst-case scenario for a penalized quadratic cost function. Through numerical analysis and three special cases, we find that the intergeneration risk-sharing is achieved in our collective hybrid pension plan effectively. And it also shows that when people live longer, postponing the retirement seems a reasonable way to alleviate the stress of the aging problem.

Stationary Points of a Shallow Neural Network with Quadratic Activations and the Global Optimality of the Gradient Descent Algorithm

We consider the problem of training a shallow neural network with quadratic activation functions and the generalization power of such trained networks. Assuming that the samples are generated by a full rank matrix [Formula: see text] of the hidden network node weights, we obtain the following results. We establish that all full-rank approximately stationary solutions of the risk minimization problem are also approximate global optimums of the risk (in-sample and population). As a consequence, we establish that, when trained on polynomially many samples, the gradient descent algorithm converges to the global optimum of the risk minimization problem regardless of the width of the network when it is initialized at some value [Formula: see text], which we compute. Furthermore, the network produced by the gradient descent has a near zero generalization error. Next, we establish that initializing the gradient descent algorithm below [Formula: see text] is easily achieved when the weights of the ground truth matrix [Formula: see text] are randomly generated and the matrix is sufficiently overparameterized. Finally, we identify a simple necessary and sufficient geometric condition on the size of the training set under which any global minimizer of the empirical risk has necessarily zero generalization error. Funding: The research of E. C. Kizildag is supported by Columbia University, with the Distinguished Postdoctoral Fellowship in Statistics. Support from the National Science Foundation [Grant DMS-2015517] is gratefully acknowledged.

Qualitative Properties of the Solution of a Conjugate Problem of Thermal Convection

The joint convection of two viscous heat-conducting liquids in a three-dimensional layer bounded by flat solid walls was studied. The upper wall is thermally insulated, and the lower wall has a non-stationary temperature field. The liquids are immiscible and separated by a flat interface with complex conjugation conditions set on it. The evolution of this system in each liquid was described by the Oberbeck–Boussinesq equations. The solution of the problem was sought for velocities that are linear in two coordinates and temperature fields that are quadratic functions of the same coordinates. Thus, the problem was reduced to a system of 10 nonlinear integro-differential equations. Its conjugate and inverse nature is determined by the four functions of time. Integral redefinition conditions were set to find them. The physical meaning of the integral conditions is the closeness of the flow. The inverse initial-boundary value problem describes convection near the temperature extremum point on the lower solid wall in a two-layer system. For small Marangoni numbers, the problem was approximated linearly (the Marangoni number is analogous to the Reynolds number in the Navier–Stokes equations). Using the obtained a priori estimates, sufficient conditions were identified for the non-stationary solution to become a stationary one over time.

Research on the frost resistance performance of fully recycled pervious concrete reinforced with fly ash and basalt fiber

In this paper, fly ash (FA) and basalt fiber (BF) were considered to solve the poor frost resistance in recycled pervious concrete (RAPC) prepared with 100 % recycled concrete aggregates (RCA), and the optimal FA replacement rate for cement and BF content were researched. The results indicate that partial replacement cement by FA not only improved the strength of RAPC but also enhanced its frost resistance. The addition of BF further improved the frost resistance of RAPC and enhanced its overall integrity after freeze-thaw cycles. When the replacement rate of FA is 6 % and the content of BF is 4 kg/m3, the maximum compressive strength of RAPC is 24.3 MPa. When the FA replacement rate was 6 % and the BF content was 6 kg/m3, the specimens exhibited a minimum loss of mass is 0.9 % after 100 freeze-thaw cycles, the highest relative dynamic modulus of elasticity (RDME) was 62.5 %, and the minimum damage variable was 34.4 %. Besides, a quadratic function that could effectively predict the relationship between freeze-thaw damage of RAPC and the number of freeze-thaw cycles was established. The use of FA can not only reduce the amount of cement, but also improve the mechanical properties and frost resistance of RAPC. The addition of BF can further improve the integrity of RAPC after freeze-thaw cycles.