Integral Form Research Articles

Closed-form expressions for magnetic polarizabilities of circular apertures excited by nonuniform magnetic fields

Abstract The problem of electromagnetic penetration through an aperture in a perfectly electrically conducting plane is a classical problem in electromagnetic theory and applications. This work aims to investigate the behavior of an electrically small aperture when excited by nonuniform magnetic fields. Based on the Taylor expansion of the exact solution in integral form of the corresponding mixed boundary problem, we derive the equivalent magnetic dipole moment of a circular aperture. Closed-form expressions of dipole moments are derived. Also, for the special case of axisymmetric excitation where the dipole moment vanishes, we develop a quadrupole model to describe the aperture behavior. The effectiveness of the obtained expressions is verified by comparisons with accurate analytical solutions in special cases and the exact solutions in integral form.

The average number of integral points on the congruent number curves

We show that the total number of non-torsion integral points on the elliptic curves ED:y2=x3−D2x, where D ranges over positive squarefree integers less than N, is O(N(log⁡N)−14+ϵ). The proof involves a discriminant-lowering procedure on integral binary quartic forms and an application of Heath-Brown's method on estimating the average size of the 2-Selmer groups of the curves in this family.

Liouville-type theorem for higher order Hardy-Hénon type systems on the sphere

In this paper, we study Liouville type theorems for the positive solutions to the following higher order Hardy-Hénon type system involving the conformal GJMS operator on the sphere Sn. In order to study this we first employ the Mobius transform to transform the above Hardy-Hénon type system on the sphere Sn into a higher order elliptic system on Rn. Then, we show that every positive solution of the higher order elliptic system on Rn is a solution to the associated integral system on Rn by using polyharmonic average and iteration arguments. We use the method of moving planes in integral form to prove that there are no positive solutions for the integral system on Rn. Finally, together with the symmetry of the sphere Sn, we obtain the Liouville type theorem of the higher order Hardy-Hénon type system involving the GJMS operator on the sphere. The results of this paper are also new even for the Lane-Emden system on the sphere.

Statistics for 3-isogeny induced Selmer groups of elliptic curves

Given a sixth power free integer a, let Ea be the elliptic curve defined by y2=x3+a. We prove explicit results for the lower density of sixth power free integers a for which the 3-isogeny induced Selmer group of Ea over Q(μ3) has dimension ≤1. The results are proven by refining the strategy of Davenport–Heilbronn, by relating the statistics for integral binary cubic forms to the statistics for 3-isogeny induced Selmer groups.

Aeroelastic Structural Analysis to Calculate Symmetrical Divergence Modes of an Aircraft Wing Using Aerodynamic Lifting Line Theory

This work explores the use of numerical matrix iteration to determine an aircraft wing's divergence speed using the aerodynamic span effects approach. The present work begins with the construction of equilibrium equations in differential and integral forms. In this paper, integral formulas are used because it serves as a convenient basis for numerical solutions of complex practical problems. Second, the straight-tapered wing is divided into a number of Multhopp stations. Subsequently, the stations' torsional influence coefficient matrix has been calculated. Third, lifting line theory was used as a suitable choice of aerodynamic theory, and the governing equations were represented in matrix form. Finally, aerodynamic span effects are taken into consideration with induction effects according to Prandtl’s lifting line theory to calculate the symmetrical divergence speed from the lowest eigenvalue of the homogenous governing integral equation. To get the solution to converge, a matrix has been iterated using the MATLAB environment. The obtained results will aid modern aircraft designers in their understanding of wing instability in steady motion.

Driving response analysis of twisted and coiled polymer actuators by considering the viscoelastic behavior of their precursor fibers

Twisted and coiled polymer actuators (TCPA) represent a type of artificial muscle predominantly composed of viscoelastic polymers in their precursor fibers. An integral form of the viscoelastic constitutive model has been developed to predict the mechanical deformation of the precursor fibers. The model successfully accounts for the first-cycle effect and the creep deformation near the glass transition temperature of the precursor fibers. Combined with the multilayer model by Gao and Wang (2024 Smart Mater. Struct. 33 045031), which predicts the effective mechanical and thermal properties of TCPA, the proposed viscoelastic constitutive model accurately predicts the thermo-mechanical response of TCPA under the heating and cooling cycle and applied loads. It is noted that the driving strain of TCPA is sensitive to the heating and cooling cycles. When the cycle duration is short, the proposed viscoelastic model can be simplified to a linear elastic model. The numerical implementation of the viscoelastic constitutive model is detailed, with validation conducted on two polymer materials, polypropylene, and polyamide 66. The proposed constitutive model can effectively predict the driving response of TCPA under complex loading conditions.

Analytical solutions for strain-driven Timoshenko nanobeam bending using generalized functions

Static nanobeam bending problems have yielded conflicting results from many researchers. Methods suggested by numerous researchers have shown inconsistencies in capturing size effects and satisfying essential and natural boundary conditions for nanobeam configurations when subjected to different types of loading. This work proposes an alternative approach using Eringen’s strain-driven integral nonlocal elasticity to arrive at analytical solutions. The theory of generalized functions and the Dirac-delta identity are used to revisit Timoshenko nanobeam modeling aspects, focusing on nanobeam elastostatics. Proposed analytical solutions for fundamental and derived quantities based on the strain-driven nonlocal model have been compared with the results of existing approaches. A crucial drawback with the strain-driven models of Eringen in both integral and differential form is its inability to satisfy equilibrium equations, which is addressed extensively in this work.

Advanced Modeling of Hydrogen Turbines Using Generalized Conformable Calculus

This article addresses critical challenges in the transition to clean energy sources by highlighting the importance of advanced mathematical modeling and computational techniques in turbine design and operation. Specifically, we extend and generalize the work of Camporeale to advance the modeling of hydrogen turbine systems. By utilizing conformable calculus, we develop dynamic equations that analyze key aspects of turbine performance, including temperature variations in turbine blades, angular velocities of rotating shafts, and mass--energy balances within the plenum and combustion chamber. Furthermore, we incorporate Kirchhoff's equation in its generalized conformable integral form, enhancing the precision of energy balance calculations and improving the representation of heat transfer processes in the combustion chamber. This methodology introduces novel perspectives in hydrogen turbine research, contributing to the advancement of sustainable and efficient technologies. Our comprehensive approach aims to provide more accurate and efficient predictions of turbine behavior, thereby impacting the design and optimization of hydrogen-based clean energy systems.

Compartmental Model for Unemployment Dynamics in Ghana using Caputo Fractional Derivative

The study develops a compartmental model for unemployment dynamics in Ghana. The compartmental model is first developed through the method of compartmental designs under certain assumptions such as, all entrants into the labor force qualify for any job, some of the employed become unemployed with time, some of the unemployed go through skill training, among others. The compartmental model is further represented in terms of integral form, and the value of the kernel is thereafter substituted as a power law correlation function to achieve a Caputo fractional derivative model. The Caputo model is analyzed to ascertain its invariant and boundedness, its fixed points and its stability. The unemployment basic reproduction number \(\Re_0\) which determines the threshold of recruitment is analyzed using the Next Generation Method Approach, and the global stability for the unemployment-free equilibrium is also analyzed through the approach of Lyapunov function. The results from the analysis show that, there is attraction of all the solution of the model in \(\mathbb{R}_+^4\) since the region is positively invariant. Again, the results indicates that, the model has two fixed point. Thus, unemployment-free equilibrium and risky-unemployment equilibrium. Notwithstanding, the analysis of both the local and global stability of the model show that, the unemployment free equilibrium is local asymptotically stable (LAS) for \(\Re_0\) <1, and global asymptotically stable (GAS). The results indicate that, the model can examine the unemployment dynamics decision making.

Sinusoidal LED light recipes can improve rocket edible biomass and reduce electricity costs in indoor growth environments.

Accumulation of edible biomass by crop plants relies on maintenance of a high photosynthetic rates across the photoperiod, with assimilation rate (A) generally responding to increasing light intensity in a hyperbolic fashion. In natural environments light fluctuates greatly over the course of the day, however in Controlled Environmental Agricultural (CEA) systems, light intensity can be supplemented or precisely controlled using LEDs to create near optimum conditions. In such indoor growth environments light is often delivered as a square wave and recommendations to horticulturalists are given in the form of Daily Light Integrals (DLI). However, this does not take into account the slow photosynthetic induction at the start of the photoperiod and the decline of A towards the end of the photoperiod, which has been demonstrated by several previous studies. Square wave light regimes therefore potentially cause suboptimal photosynthetic utilization of the applied lighting and waste electricity. Here we have adapted light recipes to gradually increase and decrease in intensity to take account of these findings. We demonstrate that, utilising a sinusoidal light regime capped at 250 μmol m-2 s-1, it is possible to increase edible biomass of rocket (by ca. 20%) compared to square wave delivered at 250 at the same DLI. Additionally, this can be achieved using less electricity (0.6%), therefore reducing energy costs and improving profitability. We suggest that capping maximum light intensity at 250 µmol m-2 s-1 improves the operating efficiency of PSII photochemistry (Fq'/Fm') also known as the photosynthetic efficiency by maintaining A later in the photoperiod. We show that a higher electron transfer rate (ETR) is maintained in these treatments over the photoperiod compared to higher light intensity caps, resulting in a greater Daily Photochemical Integral (DPI). We attribute this to less NPQ due to a greater sink capacity for the end products of electron transport, ATP and NADPH, as A is kept high for longer.

Dynamic event-triggered consensus for stochastic delay multi-agent systems under directed topology

This paper addresses the mean-square exponential consensus (MSEC) challenge in stochastic delay multi-agent systems (MASs) under directed topology. The main contributions include avoiding infinitely fast sampling and conducting a random stability analysis. First, a dynamic event-triggered mechanism (DETM) with a fixed positive lower bound for the inter-execution time is designed to prohibit Zeno behavior. The proposed control strategy does not involve the expectations of system signals. Then, sampling errors are estimated using Itô isometry, which ensures that the fixed control at each work interval is maintained with adequate feedback capacity to mitigate the effects of these errors. Furthermore, with the help of a Halanay inequality in integral form and Lyapunov theorem framework, some conditions are proposed to guarantee the consensus of stochastic delay MASs under directed topology. Finally, an example is presented to prove the effectiveness of the DETM method.

Extracting parity-violating gravitational waves from projected tidal force tensor in three dimensions

Gravitational waves (GWs) may be produced by various mechanisms in the early universe. In particular, if parity is violated, it may lead to the production of parity-violating GWs. In this paper, we focus on GWs on the scale of the large-scale structure. Since GWs induce tidal deformations of the shape of galaxies, one can extract such GW signals by observing images of galaxies in galaxy surveys. Conventionally the detection of such signals is discussed by considering the three-dimensional power spectra of the E/B-modes. Here, we develop a complementary new technique to estimate the contribution of GWs to the tidal force tensor field projected on the celestial sphere, which is a directly observable quantity. We introduce two two-dimensional vector fields constructed by taking the divergence and curl of the projected tidal field in three dimensions. Their auto-correlation functions naturally contain contributions of the scalar-type tidal field. However, we find that the divergence of the curl of the projected tidal field, which is a pseudo-scalar quantity, is free from the scalar contribution and thus enables us to extract GW signals. We also find that we can detect parity-violating signals in the GWs by observing the nonzero cross-correlation between the divergence of the projected tidal field and the curl of it. It roughly corresponds to measuring the cross-power spectrum of E and B-modes, but these are complementary to each other in the sense that our estimator can be naturally defined locally in position space. Finally we present expressions of the correlation functions in the form of Fourier integrals, and discuss the properties of the kernels specific to the GW case, which we call the overlap reduction function, borrowing the terminology used in the pulsar timing array experiments.

Mass minimization using the reaction-diffusion level set method by considering local stress constraints in an integral form

Research in topology optimization with stress constraints has shown that defining the stress constraint locally yields better performance compared to global methods. However, an examination of the formulas for local stress constraints reveals a limitation - the constraint is only satisfied at points where it is explicitly defined, failing to guarantee satisfaction across the entire design domain. To address this shortcoming, this paper proposes utilizing an integral form of the stress constraint. The integral formulation theoretically ensures that the stress constraint is satisfied over whole design domain. The objective is to minimize the mass of plane stress structures using reaction-diffusion level set method while incorporating local stress constraints in this integral form. The methodology utilizes finite element approximations for geometry and displacements, defining local stress constraints through an integral formulation. A Lagrangian function combines objective and constraint functions, with sensitivity analysis performed during optimization based on level set function changes. Structural boundaries are updated using the Hamilton-Jacobi equation. The paper presents numerical examples with varying loads and support conditions to demonstrate the effectiveness of this integral stress constraint approach. Results illustrate the capability of the proposed method to generate optimal topologies while satisfying stress constraints across the entire design space.

On the Selmer group and rank of a family of elliptic curves and curves of genus one violating the Hasse principle

We study an infinite family of j-invariant zero elliptic curves ED:y2=x3+16D and their λ-isogenous curves ED′:y2=x3−27⋅16D, where D and D′=−3D are fundamental discriminants of a specific form, and λ is an isogeny of degree 3. A result of Honda guarantees that for our discriminants D, the quadratic number field KD=Q(D) always has non-trivial 3-class group. We prove a series of results related to the set of rational points ED′(Q)∖λ(ED(Q)), and the SL(2,Z)-equivalence classes of irreducible integral binary cubic forms of discriminant D. By assuming finiteness of the Tate-Shafarevich group, we derive a parity result between the rank of ED and the rank of its 3-Selmer group, and we establish lower and upper bounds for the rank of our elliptic curves. Finally, we give explicit classes of genus-1 curves that correspond to irreducible integral binary cubic forms of discriminant D=48035713, and we show that every curve in these classes violates the Hasse Principle.

3D coupled vibrations of asymmetric nano-particle mass sensors: Two-phase integral nonlocal strain gradient model and MD simulation

Since it is possible for the nano-particle to have any asymmetric shape and may attach to every position of the resonator-based mass sensors, any type of mass eccentricity and subsequently coupled vibrations are possible. So, in the present work, the 3D coupled axial-torsional-flexural vibrations of the mass nano-sensors are investigated considering both in-plane and out-of-plane bending as well as axial and torsional vibrations. In addition, the integral form of the two-phase local/nonlocal strain gradient (LNSG), as the consistent type of the common nonlocal strain gradient (NSG) elasticity theory, is employed for the first time to consider the size effects in the mass nano-sensors. A quasi-3D coupled FE model is constructed with no shear-locking within the complicated environment of the integral LNSG. The impacts of the LNSG elasticity and diverse types of possible coupling in changing the vibrational behavior of the mass nano-sensor are scrutinized and their crucial role in modeling of the nano-sensors is indicated. Furthermore, molecular dynamic (MD) simulations are implemented to identify and analyze the coupled vibrations of a mass nano-sensor consisting of a fixed-free carbon nanotube. Comparison between the MD results with those of the present coupled FE model reveals that considering the coupling effects reduces the modeling error, especially in the cases in which the coupling influences intensify. The present work can help to provide more accurate models of mechanical resonance-based nanosensors to detect nanoparticles with different shapes, larger sizes, and high mass ratios, which can lead to achieving sensors with higher sensitivity.

Exergy Flow as a Unifying Physical Quantity in Applying Dissipative Lagrangian Fluid Mechanics to Integrated Energy Systems.

Highly integrated energy systems are on the rise due to increasing global demand. To capture the underlying physics of such interdisciplinary systems, we need a modern framework that unifies all forms of energy. Here, we apply modified Lagrangian mechanics to the description of multi-energy systems. Based on the minimum entropy production principle, we revisit fluid mechanics in the presence of both mechanical and thermal dissipations and propose using exergy flow as the unifying Lagrangian across different forms of energy. We illustrate our theoretical framework by modeling a one-dimensional system with coupled electricity and heat. We map the exergy loss rate in real space and obtain the total exergy changes. Under steady-state conditions, our theory agrees with the traditional formula but incorporates more physical considerations such as viscous dissipation. The integral form of our theory also allows us to go beyond steady-state calculations and visualize the local, time-dependent exergy flow density everywhere in the system. Expandable to a wide range of applications, our theoretical framework provides the basis for developing versatile models in integrated energy systems.

Cluster realisations of ıquantum$\imath {\rm quantum}$ groups of type AI

AbstractThe group of type is a coideal subalgebra of the quantum group , associated with the symmetric pair . In this paper, we give a cluster realisation of the algebra . Under such a realisation, we give cluster interpretations of some fundamental constructions of , including braid group symmetries, the coideal structure and the action of a Coxeter element. Along the way, we study a (rescaled) integral form of , which is compatible with our cluster realisation. We show that this integral form is invariant under braid group symmetries, and construct the Poincare‐Birkhoff‐Witt (PBW)‐bases for the integral form.

Study on Thermal Oxygen Aging Characteristics and Degradation Kinetics of PMR350 Resin.

The thermal stability and aging kinetics of polyimides have garnered significant research attention. As a newly developed class of high thermal stability polyimide, the thermal aging characteristics and degradation kinetics of phenylene-capped polyimide prepolymer (PMR350) have not yet been reported. In this article, the thermo-oxidative stability of PMR350 was investigated systematically. The thermal degradation kinetics of PMR350 resin under different atmospheres were also analyzed using the Flynn-Wall-Ozawa method, the Kissinger-Akahira-Sunose method, and the Friedman method. Thermogravimetric analysis (TGA) results revealed that the 5% thermal decomposition temperature (Td5%) of PMR350 in a nitrogen atmosphere was 29 °C higher than that in air, and the maximum thermal degradation rate was 0.0025%/°C, which is only one-seventh of that observed in air. Isothermal oxidative aging results indicated that the weight loss rate of PMR350 and the time-dependence relationship follow a first-order exponential growth function. PMR350 resin thermal decomposition reaction under air atmosphere includes one stage, with a degradation activation energy of approximately 57 kJ/mol. The reaction model g(α) fits the F2 model, and the integral form is given by g(α) = 1/(1 - α). In contrast, the thermal decomposition reaction under a nitrogen atmosphere consists of two stages, with degradation activation energies of 240 kJ/mol and 200 kJ/mol, respectively. The reaction models g(α) correspond to the A2 and D3 models, with the integral forms represented as g(α) = [-ln(1 - α)]2 and g(α) = [1 - (1 - α)1/3]2 due to the oxygen accelerating thermal degradation from multiple perspectives. Moreover, PMR350 resin maintained high hardness and modulus even after thermal aging at 350 °C for 300 h. The results indicate that the resin exhibits excellent resistance to thermal and oxygen aging. This study represents the first systematic analysis of the thermal stability characteristics of PMR350 resin, offering essential theoretical insights and data support for understanding the mechanisms of thermal stability modification in PMR350 and its engineering applications.

Some results related to common fixed point of four maps in f-metric spaces

The purpose of this paper is to present some existence and uniqueness results for some common fixed points of four mappings in F-metric spaces that satisfy the generalized conditions. Whereas, the new proposed two theorems allow to generalize and improve some theorems published in many previous papers related to fixed point in F-metric spaces. Indeed, the basics of the two new theorems proposed in this paper include a generalization and improvement of the previous well-known theorems in metric space, F-metric space and class of F-metric spaces, leading to more flexibility in their understanding and their applications. Furthermore, the proposed theorems allow overcoming the weak contraction type faced in the common fixed point theorem in the setting of F-metric spaces. Some examples are presented to prove that the proposed theorems are complementary and applicable where the existence and uniqueness of common fixed points of four mappings in F-metric spaces in the form of integrals are obtained.

Mathematical analysis of a norm-conservative numerical scheme for the Ostrovsky equation

AbstractThe target of this study is a norm-conservative scheme for the Ostrovsky equation, as its mathematical analysis has not been addressed. First, the existence and uniqueness of its numerical solutions are demonstrated. Subsequently, a convergence estimate in the two-norm is established. This, in turn, implies a convergence in the first-order Sobolev space using a supplementary sup-norm boundedness argument. Finally, this conservative scheme can be implemented in a differential form, which is considerably better than the integral form in terms of computational cost-effectiveness.