Polynomial Form Research Articles

A Long-Memory Model for Multiple Cycles with an Application to the US Stock Market

This paper proposes a long-memory model that includes multiple cycles in addition to the long-run component. Specifically, instead of a single pole or singularity in the spectrum, it allows for multiple poles and, thus, different cycles with different degrees of persistence. It also incorporates non-linear deterministic structures in the form of Chebyshev polynomials in time. Simulations are carried out to analyze the finite sample properties of the proposed test, which is shown to perform well in the case of a relatively large sample with at least 1000 observations. The model is then applied to weekly data on the S&P 500 from 1 January 1970 to 26 October 2023 as an illustration. The estimation results based on the first differenced logged values (i.e., the returns) point to the existence of three cyclical structures in the series, with lengths of approximately one month, one year, and four years, respectively, and to orders of integration in the range (0, 0.20), which implies stationary long memory in all cases.

Simultaneous fault-tolerant consensus control and disturbance suppression for multi-agent systems with polynomial form fault under switching topology

For continuous-time linear leader-follower multi-agent systems with polynomial faults, the problem of fault-tolerant state consensus under switching topologies is studied. firstly, the average dwell time method is introduced, and observers are designed for different topologies to observe faults and disturbances. Secondly, according to the observed estimated value, the corresponding fault-tolerant control consensus protocol is proposed considering the switching situation of the communication topology. Finally, both theoretical analysis and simulation examples verify that the proposed design scheme can compensate faults and disturbance rejection.

Extended Special Linear group ESL2(F) and matrix equations in SL2(F), SL2(Z) and GL2(Fp)

The problem of roots existence for different classes of matrix such as simple and semisimple matrices from SL2(F), SL2(Z) and GL2(F) are solved. For this purpose, we first introduced the concept of an extended special linear group ESL2(F), which is generalisation of the matrix group SL2(F), where F is arbitrary perfect field. The group of unimodular matrices and extended symplectic group ESp2(R) are generalised by us, their structures are found. Our criterion oriented on a general class of matrix depending of the form of minimal and characteristic polynomials, moreover a proposed criterion holds in GL2(F), where F is an arbitrary field. The method of matrix factorisation is outlined. We show that ESL2(F) is a set of all square matrix roots from SL2(F) except of that established in our root existence criterion.

Investigation of exact solutions to the (2+1)-dimensional Sakovich equation with time dependent coefficients using analytical methods

In this paper, the exact solutions of the (2 + 1)-dimensional variable coefficient Sakovich equation are investigated. Four distinct analytical methods, containing the unified method, the modified Kudryashov method, the modified G′G2 -expansion method and the improved Riccati sub-equation method, are implemented to find new exact solutions of this equation. Soliton and periodic soliton solutions, elliptic solutions, kink solutions and traveling wave solutions, which are expressed in polynomial, rational, trigonometric and hyperbolic forms, have been obtained and are graphically represented with the aim of exhibiting the structures and dynamic behaviors of the equation. Furthermore, we have discussed the influence of variable coefficients on soliton behaviors in detail.

Polynomial-based kernel reproduced gradient descent for stochastic optimization

Stochastic gradient descent (SGD) is widely applied to machine learning tasks. However, the existence of gradient variance degrades the convergence performance of SGD, which cannot be tackled well for tasks with one-time data sampling per sample. In this paper, a kernel reproduced gradient descent algorithm is proposed where the gradient of the risk function is estimated by utilizing the Reproducing Kernel Hilbert Space (RKHS) theory. Meanwhile, to ensure a low storage requirement, a kernel function linearization strategy is used to design the polynomial form of the kernel learning algorithm. It is shown that the error resulting from linearization decreases as the model parameter size increases, and the proposed polynomial-like kernel reproduced gradient descent (PKRGD) algorithm can converge faster than SGD. From experiments on models such as the latest image matching model, LightGlue, it is verified that the reproduced gradients have lower variances, and existing optimizers using the reproduced gradients can further accelerate model training.

Reentry vehicle fixed-time terminal guidance and attitude control with impact angle constraints

The motivation of this paper is to handle the terminal guidance and attitude control problem for the unpowered reentry vehicle in the diving phase, which is crucial for interception to the target. The most important objectives of this study are to intercept the maneuvering target with the miss distance and impact angle error as small as possible. First, the second-order derivatives of the aerodynamic angles are derived with the fin deflections as control input, which is objective to avoid the tracking of reentry vehicle body angle rate command in the traditional methods. Second, a novel fixed-time time-varying parameter nonsingular terminal sliding mode guidance law is proposed to improve the convergence speed of line-of-sight angle rate, which can avoid control input singularity without switching to the polynomial form when the error approaches the origin. Thirdly, a new global time-varying sliding mode is developed to design the attitude controller, with the predefined fixed convergence time and analytical solution. The system states are proved to be fixed-time convergence based on the Lyapunov stability theory. Six-degree-of-freedom flight simulations are conducted to validate the superiority of the proposed algorithm in terms of miss distance, impact angle error, intercepting time and control input energy.

Topology of quasi divisor graphs associated with non-associative algebra

The visualization of graphs representing algebraic structures has increasingly gained traction in chemical engineering research, emerging as a significant scientific challenge in contemporary studies. Due to their practical applications, graphs associated with groups and various algebraic systems have garnered significant interest in academic literature. Chemical graph theory employs graphs to illustrate molecular structures, where vertices symbolize atoms and edges represent bonds. The structure of quasi divisor graphs linked to non-associative algebra, including quasigroups and loops, presents numerous potential implications and wider applications. These applications span multiple disciplines, such as algebra, combinatorics, computer science, and network theory. Latin squares are tabular representations of quasigroups, generalizations of group structure. Numerous classifications of quasigroups exist in academic literature, such as Moufang quasigroups, Bol quasigroups, and alternative quasigroups. However, the structural characteristics of quasigroups exhibiting the weak inverse property closely resemble those of groups. The overall purpose of this study is to see the interrelationship between non-isomorphic quasi divisor graphs and a particular class of G-loops. To achieve our results, we employ a variety of methods, including a combinatorial approach, degree counting techniques, vertex and edge partitioning methods, as well as graph-theoretical tools and analytical procedures. In this paper, we compute a few topological indices based on degree, distance, and degree-distance using the structural aspects of quasi divisor graphs of orders 2φ,22ϕ,23ϕ and 4ϕ1ϕ2 where φ is a positive integer and ϕ1,ϕ2 are odd primes with ϕ1<ϕ2. This work also includes polynomial forms and their geometrical interpretations connected to linear, quadratic and higher degree topological indices of quasi divisor graphs associated with a class of non-commutative weak inverse property quasigroups.

DISPERSION OF LAMB WAVES IN MULTILAYER STRUCTURES

Low cost, the possibility of online monitoring and high sensitivity distinguish the method of structural monitoring using Lamb waves from other available methods. Structural analysis based on Lamb waves in heterogeneous materials requires fundamental knowledge of the behavior of Lamb waves in such materials. This basic knowledge is critical for signal processing in determining possible damage that can be detected by the propagating wave. Recently, Lamb wave methods have been used to simultaneously survey large areas of composite structures. However, such methods are more complex than traditional ultrasonic testing because Lamb waves have dispersive characteristics, namely, the wave speed varies depending on the frequency, modes and thickness of the plates. Experimentally measured group velocities of Lamb waves in composite materials with anisotropic characteristics do not coincide with theoretical group velocities, which are calculated using the dispersion equation of Lamb waves. This discrepancy arises because in anisotropic materials there is an angle between the direction of the group velocity and the direction of the phase velocity. This work investigates the propagation characteristics of Lamb waves in composites, focusing on group velocity and characteristic wave curves. For symmetric laminates, a robust method is proposed by imposing boundary conditions on the mid-plane and top surface to separate symmetric and antisymmetric wave modes. The dispersive and anisotropic behavior of Lamb waves in two different types of symmetrical laminates is theoretically studied in detail. The dispersion of Lamb waves was studied for 10 symmetric and asymmetric modes. It is shown that only fundamental modes are not characterized by a cutoff frequency, which indicates the interaction of fundamental modes with composite layers in the low-frequency range. A high level of group velocity dispersion was discovered for the SH0 and S0 modes. It is concluded that in isotropic laminates, dispersion is characteristic of symmetric modes. It is shown that the frequency dependence of the group velocity of Lamb waves of laminar composites can be represented in polynomial form.

Оцінка точності комп’ютерного моделювання окуло-моторної системи на основі моделей Вольтерри

Integral nonlinear models are used to create mathematical models of the human oculo-motor system (OMS). These models take into account both inertial and nonlinear properties of the objects under study. To obtain empirical data for model construction, experimental studies are conducted with OMS using «input-output» data. Visual stimuli are used as test signals, displayed on a computer monitor at various distances from the starting position, which formally corresponds to the action of step signals with varying amplitudes on the object of study. In this process, the responses of the OMS are recorded using innovative eye-tracking technology. Mathematical models in the form of Volterra series and polynomials are employed for computer modeling of the OMS. The aim of this research is to analyze the accuracy of OMS identification as multidimensional transient functions based on eye-tracking data, examining the dependency of computation errors for models of different orders on the amplitudes and quantities of the test signals used. The subject of the study includes various methods for identifying OMS, algorithms, and Python-based software tools for computing the dynamic characteristics of OMS using eye-tracking technology. The research explores identification methods: compensation, approximation, and least squares methods. The accuracy of the linear, quadratic, and cubic OMS models is evaluated. The most accurate models, constructed from real experimental data, are found to be quadratic or cubic OMS models obtained using the least squares method with three test signals.

Density, thermal expansion, dynamic viscosity of halogenated arenes

Purpose. Experimental study of the density, thermal expansion and dynamic viscosity of halogenated arenes at temperatures from 293.15 to 343.15 K and atmospheric pressure 97.992 kPa.Methods. Density was determined by the pycnometric method, viscosity by the capillary method. A quartz pycnometer type PZh2 with a nominal capacity of 10 ml and a glass capillary viscometer type VPZh-2 with a nominal capillary diameter of 0.34 mm were used. The mass of liquids was measured using an electronic scale VSL-200/0.1A with a division value of 0.0001 g. Numerical methods were used to process the experimental results.Results. Experimental data were obtained on the density, coefficient of volumetric thermal expansion and dynamic viscosity of liquid o-xylene, ethylbenzene, fluorobenzene, chlorobenzene, bromobenzene, toluene, o-fluorotoluene, mfluorotoluene, p-fluorotoluene, o-chlorotoluene, m-chlorotoluene, p-chlorotoluene, 2,4-dichlorotoluene, 2,6-dichlorotoluene. Density and dynamic viscosity data are approximated by a third-degree polynomial. Data on the coefficient of volumetric thermal expansion are obtained from approximated measured density data. The maximum estimate of the measurement error for density was 0.13 %, for dynamic viscosity – 3.5 %. The maximum calculation error for the coefficient of thermal expansion is estimated at 3 %.Conclusion. Mathematical processing of the obtained experimental data on density and dynamic viscosity made it possible to obtain analytical relations in the form of power polynomials that allow calculating the values of these quantities at any temperature from the studied temperature range and atmospheric pressure with an error not exceeding the error of experimental determination. The calculated values are in good agreement with the data provided in the reference and scientific literature. The experimental data obtained complement the information on the properties of some of the liquids under consideration, for which the dependences of density and viscosity on temperature are poorly understood. The substances under study are used in various sectors of the economy and are therefore technically important liquids. The measurement results can be used in condensed matter physics, for the analysis of liquid hydrocarbons, and can be useful in the chemical industry.

Immersion and Invariance Adaptive Control through Polynomial Adaptation

Abstract In conventional immersion and invariance (I&amp;I) adaptive control design, control parameter adaptation is typically linear with respect to the parameter-error-induced perturbation, resulting in quadratic-rate dissipation of energy associated with the off-the-manifold variable. Departing from such a convention, this article contributes a novel strategy - polynomial adaptation. As the name suggests, control parameter adaptation in this approach takes the form of a general polynomial in relation to the perturbation. Accordingly, this new design induces polynomial-rate energy dissipation, which is faster than the quadratic one in the conventional scheme, thereby enhancing closed-loop control performance. The theoretical underpinnings of the new approach are demonstrated through the design of an I&amp;I adaptive tracking control law for a general nth-order, single-input-single-output, parametrically uncertain, nonlinear system in the controllable canonical form. Additionally, a numerical study of the proposed method on the second-order forced Duffing oscillator showcases its improved transient performance in comparison to a baseline controller developed with the standard I&amp;I adaptive control technique.

Collision dynamics between breather and lump-type localized waves in the (3+1)-dimensional shallow water wave equation

In this paper, the collision dynamics of breather and lump-type localized waves in the (3+1)-dimensional shallow water wave equation are investigated in detail. Firstly, the auto-Bäcklund transformation and the linear representation of the (3+1)-dimensional shallow water wave equation are derived in virtue of the truncated Painlevé expansion method, which provide convenience in solving the (3+1)-dimensional shallow water wave equation. Secondly, based on the linear representation and the principle of linear superposition, the rational solutions in the exponential and polynomial forms are constructed. Tuning the free parameters of the rational solutions, localized waves of various patterns are obtained such as breather, lump-type localized waves and their hybrid structure. The anomalous inelastic interaction phenomenons of breather and lump-type localized waves are exhibited. Thirdly, combining the large-time behaviors of solution with the velocity relationship of localized waves, the dynamical properties and the classification of localized wave solutions are discussed in detail. Finally, we discuss the bound state of breather and lump-type localized waves under the velocity resonance condition, three different types of lump-breather molecules are displayed. The obtained results further enrich the structures and dynamical behaviors of localized waves. It is expected that the interaction phenomena taking place in the (3+1)-dimensional shallow water wave equation will be helpful in predicting or controlling some related shallow water wave phenomena.

MATHEMATICAL MODELS OF THE THERMAL PROTECTION COATING OF THE GAS GENERATOR OF THE HYDROGEN STORAGE AND SUPPLY SYSTEM

The article describes the properties of the thermal protection coating of the gas generator of the hydrogen storage and supply system by two transfer functions with Hurwitz polynomials of the third order. Obtaining such transfer functions is based on the solution of the non-stationary heat conduction equation, represented in operator form using the integral Laplace transform, and in which approximating spline functions in the form of second-order polynomials are used. The article gives the solution of the non-stationary thermal conductivity equation, which describes the thermal processes in the heat-protective coating of the gas generator of the hydrogen storage and supply system under the thermal effect of a fire. This solution comes in the operator form for the surface temperature of the heat-protective coating on the side of the gas generator wall. The peculiarity of this decision is the presence of hyperbolic functions of an irrational argument in its composition. The structural and dynamic scheme of the thermodynamic system ‘gas generator wall – heat-protective coating’ is presented, the feature of which is the presence of two entrances. The signal at the scheme’s first input reflects the thermal effect of the fire, and the signal at the second input reflects the thermal state of the gas generator cavity. An equivalent mathematical transition to the description of thermal processes in the heat-protective coating of the gas generator of the hydrogen storage and supply system was carried out. This transition happened due to the use of spline functions, which approximate the hyperbolic functions of the irrational argument and are polynomials of the second order. The article gives a verbal interpretation of the algorithm for determining the transfer functions of the heat-protective coating of the gas generator of the hydrogen storage and supply system and also an example of its implementation. Furthermore, it shows that for the given conditions of functioning of the gas generator heat-protective coating, the relative errors in approximation of hyperbolic functions by second-order polynomials do not exceed 1.7 %, and the average relative error when equivalent replacement of transfer functions does not exceed 3.8 %. Keywords: gas generator, hydrogen storage and supply system, thermal protection coating.

A new modified Bessel‐type radial basis function for meshless methods: Utilized in the analysis of free vibration in 2D functionally graded Euler–Bernoulli beams

AbstractRadial basis functions (RBFs) are extensively employed in mesh‐free methods owing to their distinct properties. This study presents a novel RBF formulation based on a modified first‐kind Bessel function, introduced for the first time. The efficacy and precision of the proposed function are assessed through an examination of the free vibrations of Euler–Bernoulli beams composed of two‐directional functionally graded materials. Longitudinal and thickness property variations are modeled in polynomial and exponential forms, respectively. The performance of the novel RBF is scrutinized under various boundary conditions (clamped, simply supported, and free), and comparative analyses are conducted against similar investigations and an RBF based on the first‐kind Bessel function. Convergence analysis of the proposed modified first‐kind Bessel function‐based RBF reveals superior convergence rates compared to the first‐kind Bessel function‐based RBF. Moreover, a comparison between results obtained from modeling using the proposed RBF and exact solutions underscores the adequacy of this approach, with a maximum discrepancy of 4.933% observed under clamped‐free boundary conditions. In essence, the findings suggest that the proposed modified first‐kind Bessel function‐based RBF holds promise for analyzing the free vibrations of functionally graded Euler–Bernoulli beams. The primary aim of this research is to introduce and validate a new RBF based on a modified first‐kind Bessel function for the analysis of free vibrations in Euler–Bernoulli beams made of two‐directional functionally graded materials. The study focuses on evaluating the performance and accuracy of this novel RBF in comparison with existing RBFs and exact solutions. By addressing the limitations of conventional RBFs and proposing an innovative approach, this research aims to enhance the accuracy and efficiency of meshless methods in structural vibration analysis.

Form factors, spectral and Källén-Lehmann representation in nonlocal quantum gravity

We discuss the conical region of convergence of exponential and asymptotically polynomial form factors and their integral representations. Then, we calculate the spectral representation of the propagator of nonlocal theories with entire form factors, in particular, of the above type. The spectral density is positive-definite and exhibits the same spectrum as the local theory. We also find that the piece of the propagator corresponding to the time-ordered two-point correlation function admits a generalization of the Källén-Lehmann representation with a standard momentum dependence and a spectral density differing from the local one only in the presence of interactions. These results are in agreement with what already known about the free theory after a field redefinition and about perturbative unitarity of the interacting theory. The spectral and Källén-Lehmann representations have the same standard local limit, which is recovered smoothly when sending the fundamental length scale ℓ* in the form factor to zero.

Effective Method for Solving Higher-Order Fractional Differential Equations Using Spline Functions

In this paper, appropriate spline functions in polynomial form are outlined and applied to solve higher-order linear fractional differential equations (H-OLFDEs). Several techniques have been proposed to solve this type of equations. A description of the projected methodology is first introduced. The method involves transforming the H-OLFDE of order α, with given initial conditions, into a system of fractional differential equations with , denoting the order of the Caputo fractional derivative for each equation i in the transformed system, where the number of equations in the resulting system is equal to that of initial conditions. The numerical results obtained using this strategy demonstrate substantial agreement with the precise analytical solutions. The results in this paper affirm the robustness and ease of application of the proposed technique.

Investigation of the accuracy of integral models of the oculo-motor system

For mathematical modeling of the human oculomotor system (OMS), integral nonlinear models are used, which simultaneously take into account the nonlinear and inertial properties of the research object. Based on the data of experimental studies of the OMS "input-output", transient and diagonal intersections of transient functions of the second and third orders are determined. To obtain experimental data, an innovative eye tracking technology is used, which allows recording eye responses to test visual stimuli. Thus test signals are displayed on the computer monitor at different distances from the starting position in the horizontal direction. The aim of the research is to study the accuracy of OMS identification using eye-tracking data by evaluating the calculation errors of multidimensional transient functions when using methods of nonlinear dynamic identification based on models in the form of Volterra series and polynomials. The object of the study is the process of nonparametric identification of the OMS based on Volterra models in the time domain. The subject of the research is algorithmic and software tools for calculating the dynamic characteristics of OMS based on eye-tracking data, analyzing the accuracy of the obtained models using two identification methods: the approximation method and the least squares method (LSM). The means of nonlinear dynamic identification of the human OMS based on Volterra series and polynomials were developed in the Python programming environment. The accuracy estimates of various OMS (linear, quadratic, and cubic) models were obtained based on data from three responses to test signals of different amplitudes. For the same test signals, the same models in the form of the Volterra series and polynomials were obtained, as these models coincide in the region of convergence of the Volterra series. The analysis of errors in the assessment of the dynamic characteristics of the OMS demonstrated that the model in the form of an integral polynomial of the second degree, which was built using the LSM based on three responses, has an accuracy twice as high as the accuracy of similar models built using the data of two responses. Thus, in further studies of the psychophysiological state of a person based on nonlinear dynamic models of the OMS according to the data of three responses, it is advisable to use a model in the form of a quadratic Volterra polynomial. Figs.: 17. Tabl. 3. Refs.: 10 titles.

Enhancing High-Alloy Steel Cutting with Abrasive Water Injection Jet (AWIJ) Technology: An Approach Using the Response Surface Methodology (RSM).

The common machining technologies for difficult-to-machine materials do not remarkably ensure acceptable efficiency and precision in bulk materials cutting. High-energy abrasive water injection jet (AWIJ) treatment can cut diverse materials, even multi-layer composites characterized by divergent properties, accurately cutting complex profiles and carrying them out in special circumstances, such as underwater locations or explosion hazard areas. This work reports research on the AWIJ machining quality performance of X22CrMoV12-1 high-alloy steel. The response surface method (RSM) was utilized in modeling. The most influencing process control parameters on cut kerf surface roughness-abrasive flow rate, pressure, and traverse speed-were tested. The result is a mathematical model of the process in the form of a three-variable polynomial. The key control parameter affecting the cut slot roughness turned out to be the traverse speed. In contrast, pressure has a less significant effect, and the abrasive mass flow rate has the slightest impact on the cut slot roughness. Under the optimal conditions determined as a result of the tests, the roughness of the intersection surface Sq does not exceed 2.3 μm. Based on the ANOVA, we confirmed that the model fits over 96% appropriately with the research outcomes. This method reduces the computations and sharply determines the optimum set of control parameters.

Efficient Optimization for Time-Constrained Encounter of Spacecraft with Multirevolution Phasing

“Spacecraft encounter” refers to a close flyby or fast proximity between two or more spacecraft in space. It involves the approach of one spacecraft to another, whether for observation, communication, capture, or interception purposes. This paper presents a novel approach to planning orbital encounter missions for spacecraft within a specified time constraint. The proposed method takes advantage of a unique encounter process where the vehicle performs only two in-plane impulses and matches the target location at the encounter position using multirevolution phasing. To ensure the minimum maneuvering fuel consumption from the selection of the encounter position and analyze the optimality of the proposed encounter maneuver, the determination method of the minimal orbital intersection distance is first given by constructing constraint equations in polynomial form. Then, the algorithm framework for determining the optimal encounter maneuvers is established. This scheme allows us to derive the optimal maneuver position, direction, and impulse magnitude, thereby creating a detailed two-layer solution framework for an arbitrary target. Concurrently, the strategy accommodates orbital phase deviations by proffering a rapid correction method that factors in the perturbation environment. To expand this optimal encounter maneuver planning method to multitarget missions, we develop a reduced-order allocation procedure for optimal pairing between vehicles and targets. Through the use of numerical simulations exemplified by one single-target and two multitarget missions, it is evident that the proposed method is both highly efficient and effective for encountering any given target.

Identification of MIMO Nonlinear Gaussian Time-Varying System Based on Multi-Dimensional Taylor Network Multi-Level Approximation

Aiming at the problems of identification difficulties and low identification accuracy in modelling and identification of multiple-input multiple-output (MIMO) nonlinear Gaussian time-varying systems, this paper proposes an identification scheme based on the step-by-step approximation of multidimensional Taylor network (MTN). The aim of this paper is to improve the modelling of complex nonlinear systems so as to improve the prediction performance and control effect of the system. Different from the traditional multidimensional Taylor network identification method, this method adopts an order-by-order approximation strategy, which seeks its parameters sequentially from the lower order to the higher order, and continuously optimises the parameter weights during the parameter seeking process. Firstly, the nonlinear function model is approximated as a polynomial form by the order-by-order Taylor expansion, and then the weight parameters of each order of the Taylor expansion are calculated and updated step by step by using the algorithm based on the Variable Forgetting Factor Recursive Least Squares (VFF-RLS) method. Through iterative optimized of these parameters, dynamic weight assignment to each order of the Taylor expansion is achieved. A parameter-identified nonlinear function model is finally obtained, which can more accurately describe the dynamic behaviour and characteristics of the system. Finally, an arithmetic simulation is carried out through an example to verify the effectiveness of the proposed method.