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Motivated by the consideration of integrable systems in three spatial dimensions in Euclidean space with integrals quadratic in the momenta we classify three-dimensional Abelian subalgebras of quadratic elements in the universal enveloping algebra of the Euclidean algebra under the assumption that the Casimir invariant p â†' â‹... l â†' vanishes in the relevant representation. We show by means of an explicit example that in the presence of magnetic field, i.e. terms linear in the momenta in the Hamiltonian, this classification allows for pairs of commuting integrals whose leading order terms cannot be written in the famous classical form of Makarov et al [17]. We consider limits simplifying the structure of the magnetic field in this example and corresponding reductions of integrals, demonstrating that singularities in the integrals may arise, forcing structural changes of the leading order terms. [ABSTRACT FROM AUTHOR]

The development of two‐dimensional (2D)/three‐dimensional (3D) switchable displays in the display market is limited due to the narrow depth of 3D display mode. In this paper, we propose a depth‐enhanced 2D/3D switchable display based on integral imaging that stacks multiple polarization switching layers and multiple polarization‐dependent liquid crystal micro‐lens arrays (LCMLAs) together. According to the change of the incident polarization state by the polarization switching layer, the polarization‐dependent LCMLA can realize the focusing effect and transmission effect respectively. Because different layers of polarization‐dependent LCMLA have different gaps (g) and the combination of different layers of polarization‐dependent LCMLA obtains different focal lengths (f), hence, multiple central depth planes can be generated to improve the depth of field (DOF) of the 3D mode. The 2D display mode is realized when all the layers of the polarization‐dependent LCMLA are in transmission effect. We experimentally developed a prototype of the proposed 2D/3D switchable display with two polarization switching layers and two polarization‐dependent LCMLAs. The results show that the DOF of the prototype is easily increased by more than two times. At the same time, the 2D image is also clearly presented in 2D mode without any distortion. [ABSTRACT FROM AUTHOR]

It is shown that in A. Einstein's 1915 paper 'Explanation of the perihelion motion of Mercury from the general theory of relativity' an error was actually incurred in the integration of the equation $$ \phi =\left[1+\upalpha \left({\upalpha}_1+{\upalpha}_2\right)\right]\underset{\upalpha_1}{\overset{\upalpha_2}{\int }}\frac{dx}{\sqrt{-\left(x-{\upalpha}_1\right)\left(x-{\upalpha}_2\right)\left(1-\upalpha x\right)}} $$, and in the result instead of $$ \phi \approx \uppi \left[1+\frac{5}{4}\upalpha \left({\upalpha}_1+{\upalpha}_2\right)\right] $$ (if we limit ourselves to first-order terms in the small quantity (α + α)) the value $$ \phi \approx \uppi \left[1+\frac{3}{4}\upalpha \left({\upalpha}_1+{\upalpha}_2\right)\right] $$ was obtained, where α and α are the inverse values of the maximum and minimum distances of Mercury from the Sun, $$ \upalpha =\frac{2G{m}_0}{c^2} $$ is the gravitational radius, G is the gravitational constant, m is the mass of the Sun, c is the velocity of light (i.e., the Chinese mathematician Hua Di was actually right). And, as a result, for the precession of the orbit of Mercury in the gravitational field of a spherical Sun in the general theory of relativity (after 100 years) one obtains not ~43', as A. Einstein obtained, but ~71.63'. Exactly this latter result is obtained by direct numerical modeling of the precession of the perihelion of Mercury's orbit in the gravitational field of the Sun within the framework of the general theory of relativity if the fitting coefficient α in the equation of motion of Mercury (not to be confused with α in the above equations) is set equal to zero. The result $$ \phi \approx \uppi \left[1+\frac{3}{4}\upalpha \left({\upalpha}_1+{\upalpha}_2\right)\right] $$, obtained by A. Einstein if it is obtained, happens only upon integration of either the equation $$ \phi =\left[1+\frac{\upalpha}{2}\left({\upalpha}_1+{\upalpha}_2\right)\right]\underset{\upalpha_1}{\overset{\upalpha_2}{\int }}\frac{dx}{\sqrt{-\left(x-{\upalpha}_1\right)\left(x-{\upalpha}_2\right)\left(1-\upalpha x\right)}} $$ (if we also limit ourselves to first-order terms in the small quantity (α + α)), i.e., in front of the integral in brackets the coefficient 1/2 should stand in front of the quantity α, or the equation $$ \phi =\left[1+\upalpha \left({\upalpha}_1+{\upalpha}_2\right)\right]\underset{\upalpha_1}{\overset{\upalpha_2}{\int }}\frac{dx}{\sqrt{-\left(x-{\upalpha}_1\right)\left(x-{\upalpha}_2\right)\left(1-\upalpha x\right)}} $$ (if we also limit ourselves to first-order terms in the small quantity (α + α)), i.e., in the denominator under the differential in the integral, inside the square root\ a plus sign should stand in front of the quantity α x. [ABSTRACT FROM AUTHOR]

Although Selberg-type single positive definite symmetric matrices gamma and beta integrals have been evaluated by several authors, see e.g., Askey and Richards [1], Gupta and Kabe [2, 4], Mathai [8], and elsewhere in the vast multivariate statistical analysis literature. However, several other types of Selberg-type integrals appear to have been neglected in the literature. Thus e.g., Selberg-type integrals associated with inverse Wishart densities, inverse multivariate beta densities, their noncentral counterparts, etc, have not been explored as yet. The present paper records Selberg-type generalized quadratic forms gamma and beta integrals. Our methodology is based on hypercomplex (HC) multivariate normal distribution theory, Kabe [6]. [ABSTRACT FROM AUTHOR]

In Fritz John’s famous paper (1979), he discovered that for the wave equation □ u = ∣ u ∣ p , where 1 < p < 1 + 2 and □ denoting the d’Alembertian, there is no global solution for any nontrivial and compactly supported initial data. This paper is intended to simplify his proof by applying a Gronwall’s type inequality. [ABSTRACT FROM AUTHOR]

In decision problems involving two dimensions (like several agents in uncertainty) the properties of expected utility ensure that the result of a two-stepped procedure evaluation does not depend on the order with which the aggregations of local evaluations are performed (e.g., agents first, uncertainty next, or the converse). We say that the aggregations on each dimension commute. In a previous conference paper, Ben Amor, Essghaier and Fargier have shown that this property holds when using pessimistic possibilistic integrals on each dimension, or optimistic ones, while it fails when using a pessimistic possibilistic integral on one dimension and an optimistic one on the other. This paper studies and completely solves this problem when more general Sugeno integrals are used in place of possibilistic integrals, leading to double Sugeno integrals. The results show that there are capacities other than possibility and necessity measures that ensure commutation of Sugeno integrals. Moreover, the relationship between two-dimensional capacities and the commutation property for their projections is investigated. [ABSTRACT FROM AUTHOR]

The paper is devoted to comparison of experimental data with theoretical predictions concerning the dependence of the current of accelerated ions on the operating voltage of a Hall thruster with an anode layer. The error made in the paper published by the authors in Plasma Phys. Rep. 40, 229 (2014) occurred because of a misprint in the Encyclopedia of Low-Temperature Plasma. In the present paper, this error is corrected. It is shown that the simple model proposed in the above-mentioned paper is in qualitative and quantitative agreement with experimental results. [ABSTRACT FROM AUTHOR]

A high‐resolution integral imaging 3D display system without color moiré pattern and graininess is proposed. The system consists of a 2D display, a micro‐lens array, a holographic diffuser, and a transmissive mirror device. The experimental results prove the feasibility of the proposed system. [ABSTRACT FROM AUTHOR]

The principal tool of computation used in this paper is classical Gaussian quadrature on the interval [0,1], which happens to be particularly effective here. Explicit expressions are found for the derivatives of the Ramanujan integral in question, and it is proved that the latter is completely monotone on (0,\infty). As a byproduct, known series expansions for incomplete gamma functions are examined with regard to their convergence properties. The paper also pays attention to another famous integral, the Euler integral — better known as the gamma function — revitalizing a largely neglected part of the function, the part corresponding to negative values of the argument, which plays a prominent role in our work. [ABSTRACT FROM AUTHOR]

This paper concentrates on presenting a reliable procedure to compute the stabilizing region of fractional‐order proportional integral derivative (FOPID) controllers for interval fractional‐order plants having an interval time delay. An interval fractional‐order plant is defined as a fractional‐order transfer function whose denominator and numerator coefficients are all uncertain and lie in specified intervals. Also, an interval time delay points to a delay term whose value varies in a specific interval. The D‐decomposition technique and the value set concept are employed to determine the stabilizing region of FOPID controllers. In this study, first, a theorem is presented to compute the boundary of the value sets of systems having interval time day. Then, a lemma is provided for robust stability analysis of the given closed‐loop control system. For a convenient use of the paper results, an algorithm is proposed to solve the problem of robustly stabilizing interval fractional‐order plants with an interval time delay using FOPID controllers. Finally, four examples are provided to illustrate the proposed procedure. [ABSTRACT FROM AUTHOR]

In this paper we prove a global spacetime bound for the quintic, nonlinear wave equation in three dimensions. This bound depends on the L_{t}^{\infty } L_{x}^{2} and L_{t}^{\infty } \dot {H}^{2} norms of the solution to the quintic problem. The main motivation for this paper is the use of an interaction Morawetz estimate for the nonlinear wave equation. [ABSTRACT FROM AUTHOR]

Abstract: In this note, we point out an error in the recently published article [R.K. Mohanty, M.K. Jain, D. Dhall, A cubic spline approximation and application of TAGE iterative method for the solution of two-point boundary value problems with forcing function in integral form, Appl. Math. Model. 35 (2011) 3036–3047] and then correct it. [Copyright &y& Elsevier]

We give a correction to the proof of Theorem 2.3 in the paper of E. M. Elabbasy and T. S. Hassan, Serdica Math. J. 34 (2008), 531–542. [ABSTRACT FROM AUTHOR]

This paper introduces a Linear Quadratic Gaussian (LQG) controller for a Single-Ended Primary Inductor Converter (SEPIC). The LQG design is based on merging an integral Linear Quadratic Regulator (LQR) with an offline Kalman Filter (commonly referred to as a Linear Quadratic Estimator (LQE)). The robustness of the LQG controller is guaranteed based on the separation principle. This manuscript addresses the need to use observer-based systems for the fourth-order SEPIC, which needs a sensor reduction as an essential requirement. This paper provides a comprehensive, yet systematic, approach to designing the LQG system. The work validates the convergences of the states in an LQG system to an actual value. Furthermore, it compares the performance of an LQG system with a benchmark Type-II industrial controller by means of a simulation of the switched converter model in the Simulink/MATLAB 2023a environment. [ABSTRACT FROM AUTHOR]

Helicoidal surfaces of constant mean curvature were fully described by do Carmo and Dajczer. However, the obtained parameterizations are given in terms of somewhat complicated integrals, and as a consequence, not many examples of such surfaces are visualized. In this paper, by using these methods in some particular cases, we provide several interesting visualizations involving these surfaces, mostly as an isometric deformation of a rotational surface. We also give interpretations of some older results involving helicoidal surfaces, motivated by the work carried out by Malkowsky and Veličković. All of the graphics in this paper were created in Wolfram Mathematica. [ABSTRACT FROM AUTHOR]

Monogenity is a classical area of algebraic number theory that continues to be actively researched. This paper collects the results obtained over the past few years in this area. Several of the listed results were presented at a series of online conferences titled "Monogenity and Power Integral Bases". We also give a collection of the most important methods used in several of these papers. A list of open problems for further research is also given. [ABSTRACT FROM AUTHOR]

In this note, two interesting integrals involving the product of two generalized hypergeometric functions have been evaluated in terms of gamma function. The results are derived with the help of a known integral involving hypergeometric function recorded in the paper of Lavoie and Trottlier. We also give several interesting special cases of our main findings. [ABSTRACT FROM AUTHOR]

This paper is the second in a series of four papers aiming to describe the (almost integral) Chow ring of \overline {\mathcal {M}}_3, the moduli stack of stable curves of genus 3. In this paper, we introduce the moduli stack \widetilde {\mathcal {H}}_g^r of hyperelliptic A_r-stable curves and generalize the theory of hyperelliptic stable curves to hyperelliptic A_r-stable curves. In particular, we prove that \widetilde {\mathcal {H}}_g^r is a smooth algebraic stack that can be described using cyclic covers of twisted curves of genus 0 and it embeds in \widetilde {\mathcal M}_g^r (the moduli stack of A_r-stable curves) as the closure of the moduli stack of smooth hyperelliptic curves. [ABSTRACT FROM AUTHOR]

This paper introduces new versions of Hermite–Hadamard, midpoint- and trapezoid-type inequalities involving fractional integral operators with exponential kernels. We explore these inequalities for differentiable convex functions and demonstrate their connections with classical integrals. This paper validates the derived inequalities through a numerical example with graphical representations and provides some practical applications, highlighting their relevance to special means. This study presents novel results, offering new insights into classical integrals as the fractional order β approaches 1, in addition to the fractional integrals we examined. [ABSTRACT FROM AUTHOR]

G. Thompson [J. Math. Phys. 25, 3474 (1984)], by way of example of a general method, examined the existence of integrals of the form I = (yp[sub x] - xp[sub y])³ + A (x,y)p[sub x] + B (x,y)p[sub y], for the Hamiltonian H = ½ (p²[sub x] + p²[sub y]) + V(x,y). The treatment was not complete. A complete solution is given here as well as indications for further development to integrals of higher order in the momenta. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the weak integral closure of a filtration relative to a module and the asymptotic prime divisors of a filtration relative to a module. Based on these concepts, we prove some new results. One of the most important points of this paper is to find the effect of contraction, and extension on the integral closure of a filtration relative to a module. Finally, the theorems we prove enable us to characterize the asymptotic prime divisors of contraction of a filtration f relative to a module M according to the asymptotic prime divisors of the filtration f relative to a module M. [ABSTRACT FROM AUTHOR]

This paper concerns a classical subject regarding the structural properties of abelian number fields. The primitive elements, integral bases, conductors and discriminants are important for studying abelian number fields, and this paper emphasizes that they are closely related to the associated Dirichlet characters. Then by evaluating the Gauss sums, we can give explicit forms of abelian fields of small degree. [ABSTRACT FROM AUTHOR]

Given a construction f on groups, we say that a group G is f -realisable if there is a group H such that G ≅ f (H) , and completely f -realisable if there is a group H such that G ≅ f (H) and every subgroup of G is isomorphic to f (H 1) for some subgroup H 1 of H and vice versa. In this paper, we determine completely Aut -realisable groups. We also study f -realisable groups for f = Z , F , M , D , Φ , where Z (H) , F (H) , M (H) , D (H) and Φ (H) denote the center, the Fitting subgroup, the Chermak–Delgado subgroup, the derived subgroup and the Frattini subgroup of the group H , respectively. [ABSTRACT FROM AUTHOR]

The objective of this paper is to characterize the structure of the set \Theta for a continuous ergodic upper probability \mathbb {V}=\sup _{P\in \Theta }P \Theta contains a finite number of ergodic probabilities; Any invariant probability in \Theta is a convex combination of those ergodic ones in \Theta; Any probability in \Theta coincides with an invariant one in \Theta on the invariant \sigma-algebra. The last property has already been obtained in Cerreia-Vioglio, Maccheroni, and Marinacci [Proc. Amer. Math. Soc. 144 (2016), pp. 3381–3396], which first studied the ergodicity of such capacities. As an application of the characterization, we prove an ergodicity result, which improves the result of Cerreia-Vioglio, Maccheroni, and Marinacci [Proc. Amer. Math. Soc. 144 (2016), pp. 3381–3396] in the sense that the limit of the time means of \xi is bounded by the upper expectation \sup _{P\in \Theta }E_P[\xi ], instead of the Choquet integral. Generally, the former is strictly smaller. [ABSTRACT FROM AUTHOR]

In this paper we consider various subclasses of normalized, analytic functions defined in the open unit disk A = {z = C : z < 1} in the complex plane C and study the Hardy space of the functions in these subclasses. This study provides an analysis of the growth of these functions near the boundary of the open unit disk and the Taylor's coefficients of them. The study is carried out using the methods of integral means and subordination of analytic functions. Determination of explicit indices of the Hardy space and order of the growth rate of the Taylor coefficient of these functions are important results here. The novelty of the work here is an attempt to extend the study of the above mentioned features for functions in standard subclasses of analytic univalent functions which were not considered by researchers in the past. [ABSTRACT FROM AUTHOR]

Purpose: This paper aims to propose and verify an improved cascade active disturbance rejection control (ADRC) scheme based on output redefinition for hypersonic vehicles (HSVs) with nonminimum phase characteristic and model uncertainties. Design/methodology/approach: To handle the nonminimum phase characteristic, a tuning factor stabilizing internal dynamics is introduced to redefine the system output states; its effective range is determined by analyzing Byrnes–Isidori normalized form of the redefined system. The extended state observers (ESOs) are used to estimate the uncertainties, which include matched and mismatched items in the system. The controller compensates observations in real time and appends integral terms to improve robustness against the estimation errors of ESOs. Findings: Theoretical and simulation results show that the stability of internal dynamics is guaranteed by the tuning factor and the tracking errors of external commands are globally asymptotically stable. Practical implications: The control scheme in this paper is expected to generate a reliable way for dealing with nonminimum phase characteristic and model uncertainties of HSVs. Originality/value: In the framework of ADRC, a concise form of redefined outputs is proposed, in which the tuning factor performs a decisive role in stabilizing the internal dynamics of HSVs. By introducing an integral term into the cascade ADRC scheme, the compensation accuracy of matched and mismatched disturbances is improved. [ABSTRACT FROM AUTHOR]

This paper deals with the research of solutions of bounded variation (BV) to evolution inclusion coupled with a time and state dependent maximal monotone operator. Different problems are studied: existence of solutions, unicity of the solution, existence of periodic and bounded variation right continuous (BVRC) solutions. Second-order evolution inclusions and fractional (Caputo and Riemann–Liouville) differential inclusions are also considered. A result of the Skorohod problem driven by a time- and space-dependent operator under rough signal and a Volterra integral perturbation in the BRC setting is given. The paper finishes with some results for fractional differential inclusions under rough signals and Young integrals. Many of the given results are novel. [ABSTRACT FROM AUTHOR]

In this paper, we were concerned with the global behavior of positive solutions for third-order semipositone problems with an integral boundary condition y ‴ + β y ″ + α y ′ + λ f (t , y) = 0 , t ∈ (0 , 1) , y (0) = y ′ (0) = 0 , y (1) = χ ∫ 0 1 y (s) d s , where α ∈ (0 , ∞) and β ∈ (− ∞ , ∞) are two constants, λ , χ are two positive parameters, and f ∈ C ([ 0 , 1 ] × [ 0 , ∞) , R) with f (t , 0) < 0 . Our analysis mainly relied on the bifurcation theory. In this paper, we were concerned with the global behavior of positive solutions for third-order semipositone problems with an integral boundary condition where and are two constants, are two positive parameters, and with . Our analysis mainly relied on the bifurcation theory. [ABSTRACT FROM AUTHOR]

Mathematics, both pure and applied, plays a crucial role in shaping societies and driving advancements in various fields. This research paper explores the impact of pure and applied mathematics on Nigerian society, examining their individual contributions and the synergy that emerges from their interplay. The introduction highlights the importance of mathematics in general and provides a context for understanding its significance in the Nigerian context. The subsequent sections delve into the specific impacts of pure and applied mathematics, focusing on their contributions to Nigerian society. The section on pure mathematics discusses its role in fostering critical thinking, logical reasoning, and problem-solving skills, while also showcasing the accomplishments of Nigerian mathematicians in the global mathematical community. The section on applied mathematics explores its applications in sectors such as finance, telecommunications, and engineering, and its contribution to addressing societal challenges and promoting economic development in Nigeria. The paper further investigates the synergy between pure and applied mathematics, illustrating how abstract mathematical concepts often find unexpected practical applications and how real-world challenges drive theoretical advancements. It emphasizes the collaborative efforts between researchers in both fields, the benefits of interdisciplinary perspectives, and the impact on mathematical education and pedagogy. Lastly, the paper addresses the challenges and future directions of pure and applied mathematics in Nigeria, focusing on infrastructure limitations, shortage of skilled mathematicians, integration of mathematics in industry, and the promotion of women's participation. It proposes strategies such as investment in research infrastructure, strengthening mathematics education, fostering interdisciplinary collaboration, and promoting public awareness and engagement. By addressing these challenges and charting future directions, Nigeria can harness the full potential of mathematics as a transformative force in society. [ABSTRACT FROM AUTHOR]

This study explored the unique integration of mathematics tutors in teaching high school mathematics in a Virtual mathematics school (VMS). The tutors were three excelling STEM students who did not have any formal preparation for teaching before their work in the VMS. The goal of the study presented in this paper was to design a model of proficiency for these tutors. The model was designed using multiple case study methodology by tracking the learning through teaching of the tutors. Activity theory (Leontiev, 1978) framed the design of the study and led to a "goal-action" model of the tutors' proficiency in the VMS. The goals part of the model is rooted in the construct of students' mathematical potential (Leikin, 2021), and the action part was unraveled using the construct of a Teaching Triad (Jaworski in For the Learning of Mathematics, 12(1), 8–14, 1992). In this paper, we illustrate how our model can be used to analyze changes in the tutors' proficiency, and how it can be applied to descriptive, explanatory, and analytical power. [ABSTRACT FROM AUTHOR]

For traditional Fourier transform (FTS), its integral sampling usually meets the Spectral Modulation Transfer Function (SMTF) criterion. However, for bandpass-sampling Fourier transform spectroscopy (BPS-FTS), based on our analysis, the integral sampling condition derived from the Spectral Modulation Transfer Function (SMTF) is excessively stringent. In other words, the interval of the integral sampling time that fulfills the tolerance requirements for the reconstructed spectrum is very narrow. There are numerous integration sampling time intervals outside this range that still meet the tolerance requirements for the reconstructed spectrum. In this paper, through theoretical modeling, we propose a method based on average |SMTF| as the selection criterion for the integration sampling time. Through simulation analysis, it is evident that the intervals and range of the integral sampling time obtained via this method are more accurate, ensuring the tolerance requirements of the reconstructed spectrum. Under these intervals, when conducting integral sampling on the interferogram, the spectral deviation of the reconstructed spectrum is minimal, and the Spectral Correlation Mapper (SCM) is nearly equal to one. This indicates that compared with the SMTF criterion in traditional FTS, this method is more suitable for the characteristics of BPS-FTS. The analysis in this paper can provide theoretical and simulation support for the implementation of BPS-FTS. [ABSTRACT FROM AUTHOR]

Circulating fluidized bed (CFB) boilers are widely used in industrial production due to their high combustion efficiency, low pollutant emissions and wide load-adjustment range. However, the water-level-control system of a CFB boiler exhibits time-varying behavior and nonlinearity, which affect the control performance of the industrial system. This paper proposes a novel data-driven adaptive integral sliding-mode control (ISMC) method for the CFB control system with external disturbances. Firstly, the scheme designs a discrete ISMC law based on the full-format dynamic linearization (FFDL) data model, which is equivalent to a nonlinear system. Furthermore, a new reaching law is proposed to quickly drive the system state onto the sliding-mode surface. The improved ISMC control scheme only utilizes the input–output data during the design process and does not require model information. After theoretically verifying the stability of the method proposed in this paper, it is further applied in MIMO systems. Finally, the control and practical effects of this method are evaluated by using the DHX25-1.25 CFB boiler installed in the special-equipment testing center. The experimental results show that, compared with the traditional sliding-mode control (SMC) and model-free adaptive-control (MFAC) methods, the improved control method can quickly track the given signal and exhibit resistance to noise interference. Furthermore, it can rapidly respond to changes in the working conditions of the CFB system. [ABSTRACT FROM AUTHOR]

This paper deals with a weighting technique for a class of multiple cost optimization problems with interval values. More specifically, we introduce a multiobjective interval-valued controlled model and investigate it by applying the weighting method. In this regard, several characterization theorems are derived. Moreover, a numerical example is formulated. Based on the provided illustrative example and performing a comparative analysis of the results obtained using the weighting technique versus traditional optimization methods, we can easily conclude the effectiveness of the weighting technique in solving multiple cost optimization problems, that is, the conversion of a vector problem to a scalar one. [ABSTRACT FROM AUTHOR]

In this paper, specific Lp estimates for generalized Marcinkiewicz operators correlated to surfaces of revolution are proved. These estimates and the extrapolation procedure of Yano are employed to confirm the Lp boundedness of the above-mentioned integrals under weaker assumptions on the singular kernels. Our findings generalize and improve several known results. [ABSTRACT FROM AUTHOR]

This paper proposes a novel control methodology to achieve fixed-time stabilisation for a class of three-order nonlinear systems based on fast nonsingular terminal sliding mode control. In this research, based on the combination of integral fast nonsingular terminal sliding mode (IFNTSM) surface and fixed-time control theory, a novel backstepping fast nonsingular terminal sliding mode (FNTSM) controller is being studied. The interests of this method are that the presented controller can not only maintain the characteristics of integral fast nonsingular terminal sliding mode control (IFNTSMC) but also has the advantages of fixed time convergence and fast transient response. Piecewise function is introduced to solve the singularity problem caused by negative exponential term. Our control strategy achieves the stabilisation performance within finite convergent time being independent of initial condition. The control scheme is applied to the gyros system with chaotic state and suppress its chaotic phenomenon. The theoretical results are supported by simulation results. [ABSTRACT FROM AUTHOR]

It is well known that investigation on exact solutions of nonlinear fractional partial differential equations (PDEs) is a very difficult work compared with integer‐order nonlinear PDEs. In this paper, based on the separation method of semifixed variables and integral bifurcation method, a combinational method is proposed. By using this new method, a class of generalized time‐fractional thin‐film equations are studied. Under two kinds of definitions of fractional derivatives, exact solutions of two generalized time‐fractional thin‐film equations are investigated respectively. Different kinds of exact solutions are obtained and their dynamic properties are discussed. Compared to the results in the existing references, the types of solutions obtained in this paper are abundant and very different from those in the existing references. Investigation shows that the solutions of the model defined by Riemann–Liouville differential operator converge faster than those defined by Caputo differential operator. It is also found that the profiles of some solutions are very similar to solitons, but they are not true soliton solutions. In order to visually show the dynamic properties of these solutions, the profiles of some representative exact solutions are illustrated by 3D graphs. [ABSTRACT FROM AUTHOR]

This paper is the fourth in a series of four papers aiming to describe the (almost integral) Chow ring of M ̄ 3 , the moduli stack of stable curves of genus 3. In this paper, we finally compute the Chow ring of M ̄ 3 with Z ⁢ [ 1 / 6 ] -coefficients. [ABSTRACT FROM AUTHOR]

In this paper, we propose a fourth-order compact discretization method for solving a secondorder boundary value problem governed by the nonlinear Fredholm integro-differential equations. Using an efficient approximate polynomial, a compact numerical integration method is first designed. Then by applying the derived numerical integration formulas, the original problem is converted into a nonlinear system of algebraic equations. It is shown that the proposed method is easy to implement and has the third order of accuracy in the infinity norm. Some numerical examples are presented to demonstrate its approximation accuracy and computational efficiency, as well as to compare the derived results with those obtained in the literature. [ABSTRACT FROM AUTHOR]

In 2006, Alexander proved a result that implies for a Weingarten surface Κ1/Κ2 = c > 0, if n is the number of times a closed geodesic winds around the axis of rotation and m is the number of times the geodesic oscillates about the equator, then n/m ∈ in [1, 1/c) when 0< c <1 and n/m ∈ in (1/surd c, 1] when c >1. In this paper, we present another proof of Alexander's result for the Weingarten surfaces Κ1/Κ2 =c >0 that is simpler and more direct. Our approach uses sharp estimates of certain improper integrals to obtain the intervals for permissible ratios n/m. We numerically compute a number of closed geodesics for various combinations of (c, n,m) to illustrate the variety of patterns that are possible. [ABSTRACT FROM AUTHOR]

Numerous logarithmic integrals have been extensively documented in the literature. This paper presents an algorithmic evaluation of a specific class of these integrals. Our systematic approach, rooted in logarithmic principles, enables us to extend our findings to other cases within this family of integrals. Furthermore, we explore special cases derived from our main results, thereby enhancing the applicability and significance of our work for a wider audience of researchers. [ABSTRACT FROM AUTHOR]

In this paper, we study a quantum difference coupled impulsive system with respect to another function. Some quantum derivative and integral asymmetric graphs with respect to another function are shown to illustrate the behavior of parameters. Existence and uniqueness results are established via Banach contraction mapping principle and Leray–Schauder alternative. Examples illustrating the obtained results are also included. Our results are new and significantly contribute to the literature to this new subject on quantum calculus on finite intervals with respect to another function. [ABSTRACT FROM AUTHOR]