Search Results (38)

Over-the-air federated learning (Air-FL) has emerged as a promising paradigm that integrates communication and learning, which offers significant potential to enhance model training efficiency and optimize communication resource utilization. This paper addresses the challenge of interference management in multi-cell Air-FL systems, focusing on parallel multi-task scenarios where each cell independently executes distinct training tasks. We begin by analyzing the impact of aggregation errors on local model performance within each cell, aiming to minimize the cumulative optimality gap across all cells. To this end, we formulate an optimization framework that jointly optimizes device transmit power and denoising factors. Leveraging the Pareto boundary theory, we design a centralized optimization scheme that characterizes the trade-offs in system performance. Building upon this, we propose a distributed power control optimization scheme based on interference temperature (IT). This approach decomposes the globally coupled problem into locally solvable subproblems, thereby enabling each cell to adjust its transmit power independently using only local channel state information (CSI). To tackle the non-convexity inherent in these subproblems, we first transform them into convex problems and then develop an analytical solution framework grounded in Lagrangian duality theory. Coupled with a dynamic IT update mechanism, our method iteratively approximates the Pareto optimal boundary. The simulation results demonstrate that the proposed scheme outperforms baseline methods in terms of training convergence speed, cross-cell performance balance, and test accuracy. Moreover, it achieves stable convergence within a limited number of iterations, which validates its practicality and effectiveness in multi-task edge intelligence systems. Full article

This paper develops a nonlinear optimization model for the optimal allocation of labor and investment resources in a three-sector open economy. The model is based on the Cobb–Douglas production function and incorporates sectoral interdependencies, capital depreciation, trade balances, and import quotas. The resource allocation problem is formalized as a constrained optimization task, solved analytically using the Lagrange multipliers method and numerically via the golden section search. The model is calibrated using real statistical data from Kazakhstan (2010–2022), an open resource-exporting economy. The results identify structural thresholds that define balanced growth conditions and resource-efficient configurations. Compared to existing studies, the proposed model uniquely integrates external trade constraints with analytical solvability, filling a methodological gap in the literature. The developed framework is suitable for medium-term planning under stable external conditions and enables sensitivity analysis under alternative scenarios such as sanctions or price shocks. Limitations include the assumption of stationarity and the absence of dynamic or stochastic features. Future research will focus on dynamic extensions and applications in other open economies. Full article

This study introduces a simple model, which can be used to examine the influence of reputation on expected income achieved within the Iterated Prisoner’s Dilemma (IPD) game framework. The research explores how different reputation distributions among society members impact overall outcomes by modeling a society of agents, each characterized by a reputation score that dictates their likelihood of cooperation. Due to the simplicity of the model, we can analytically determine the expected incomes based on the distribution of agents’ reputations and model parameters. The results show that a higher reputation generally leads to greater expected income, thereby promoting cooperation over defection. However, in some cases, where there are more defecting individuals, the expected income reaches the maximum for agents with an average reputation, and then decreases for individuals who cooperate more. Various scenarios, including uniform, increasing, and decreasing reputation distributions, are analyzed to understand their effects on the promoted interaction strategy. Finally, we outline future extensions of the model and potential research directions, including the exploration of alternative reputation distributions, variable interaction parameters, and different payoff structures in the dilemma games. Full article

Volatility recovery is of paramount importance in contemporary finance. Volatility levels are heavily used in risk and portfolio management. We employ the Hull–White one- and two-factor models to describe the market condition. We computationally recover the volatility term structure as a piecewise-linear function of time. For every maturity, a cost functional, defined as the squared differences between theoretical and market prices, is minimized and the respective linear part is reconstructed. On the last time steps, before each maturity, the derivative price is decomposed in order to make the minimization problem analytically solvable. The procedure works fast since only scalar values are obtained on each minimization. However, the predictor–corrector nature of the algorithm allows for the precise recovery of very complex volatility functions. An implicit scheme is used to solve the PDEs on bounded domains. The computational simulations with artificial and real data show that the proposed algorithm is stable, accurate and efficient. Full article

This paper presents a methodology for accurately incorporating the nonlinearity of boundary conditions (BCs) into the mode shapes, natural frequencies, and dynamic behaviour of analytical beam models. Such models have received renewed interest in recent years as a result of their successful implementation in state-of-the-art multiphysics problems. To address the need for this boundary nonlinearity to be more completely captured in the equations of motion, a nonlinear algebra expansion of the classical linear approach for developing solvability conditions for natural frequencies and mode shapes is presented. The method is applicable to any BC that can be accurately represented in polynomial form, either explicitly or through the application of a Taylor expansion; this is the only assumption made in removing the need for the use of analytical approximations of the dynamics themselves. By reducing the BCs of the beam to a system of polynomials, it is possible to utilise the tensor resultant to develop these solvability conditions analogous to the conditions placed on the matrix determinant in linear, classical cases. The approach is first derived for a general set of nonlinear BCs before being applied to two example systems to investigate the importance of including nonlinear tip behaviour in the BCs to accurately predict the system response. In the first, a theoretical, symmetric system, in which a beam is supported by nonlinear springs, is used to explore both the applicability of the methodology and the improvements it can make to the accuracy of the model. Then, the more practical example of a cantilever beam with repulsive magnetic interaction at the tip is used to more explicitly assess the importance of properly incorporating boundary nonlinearity into multiphysics problems. Full article

This paper investigates a high-order numerical method based on a spatial compact exponential scheme for solving the time-fractional Black–Scholes model. Firstly, the original time-fractional Black–Scholes model is converted into an equivalent time-fractional advection–diffusion reaction model by means of a variable transformation technique. Secondly, a novel high-order numerical method is constructed with $(2-\alpha )$ accuracy in time and fourth-order accuracy in space based on a spatial compact exponential scheme, where $\alpha$ is the fractional derivative. The uniqueness of solvability of the derived numerical method is rigorously discussed. Thirdly, the unconditional stability and convergence of the derived numerical method are rigorously analyzed using the Fourier analysis technique. Finally, numerical examples are presented to test the effectiveness of the derived numerical method. The proposed numerical method is also applied to solve the time-fractional Black–Scholes model, whose exact analytical solution is unknown; numerical results are illustrated graphically. Full article

Within the framework of the extended Einstein–aether–axion theory, we studied the model of a two-level aetheric control over the evolution of a spatially isotropic homogeneous Universe filled with axionic dark matter. Two guiding functions are introduced, which depend on the expansion scalar of the aether flow being equal to the tripled Hubble function. The guiding function of the first type enters the aetheric effective metric, which modifies the kinetic term of the axionic system; the guiding function of the second type predetermines the structure of the potential axion field. We obtained new exact solutions to the total set of master equations in the model (with and without cosmological constant), and studied four analytically solvable submodels in detail, for which both guiding functions are reconstructed and illustrations of their behavior are presented. Full article

This paper deals with an initial-boundary value problem modeling the unidirectional pressure-driven flow of a second grade fluid in a plane channel with impermeable solid walls. On the channel walls, Navier-type slip boundary conditions are stated. Our aim is to investigate the well-posedness of this problem and obtain its analytical solution under weak regularity requirements on a function describing the velocity distribution at initial time. In order to overcome difficulties related to finding classical solutions, we propose the concept of a generalized solution that is defined as the limit of a uniformly convergent sequence of classical solutions with vanishing perturbations in the initial data. We prove the unique solvability of the problem under consideration in the class of generalized solutions. The main ingredients of our proof are a generalized Abel criterion for uniform convergence of function series and the use of an orthonormal basis consisting of eigenfunctions of the related Sturm–Liouville problem. As a result, explicit expressions for the flow velocity and the pressure in the channel are established. The constructed analytical solutions favor a better understanding of the qualitative features of time-dependent flows of polymer fluids and can be applied to the verification of relevant numerical, asymptotic, and approximate analytical methods. Full article

In the present research, we establish an effective method for determining the time-fractional coupled Korteweg–de Vries (KdV) equation’s approximate solution employing the fractional derivatives of Caputo–Fabrizio and Atangana–Baleanu. KdV models are crucial because they can accurately represent a variety of physical problems, including thin-film flows and waves on shallow water surfaces. Some theoretical physical features of quantum mechanics are also explained by the KdV model. Many investigations have been conducted on this precisely solvable model. Numerous academics have proposed new applications for the generation of acoustic waves in plasma from ions and crystal lattices. Adomian decomposition and natural transform decomposition techniques are combined in the natural decomposition method (NDM). We first apply the natural transform to examine the fractional order and obtain a recurrence relation. Second, we use the Adomian decomposition approach to the recurrence relation, and then, using successive iterations and the initial conditions, we can establish the series solution. We note that the proposed fractional model is highly accurate and valid when using this technique. The numerical outcomes demonstrate that only a small number of terms are required to arrive at an approximation that is exact, efficient, and trustworthy. Two examples are given to illustrate how the technique performs. Tables and 3D graphs display the best current numerical and analytical results. The suggested method provides a series form solution, which makes it quite easy to understand the behavior of the fractional models. Full article

Although originally developed to describe the magnetic behavior of matter, the Ising model represents one of the most widely used physical models, with applications in almost all scientific areas. Even after 100 years, the model still poses challenges and is the subject of active research. In this work, we address the question of whether it is possible to describe the free energy A of a finite-size 2D-Ising model of arbitrary size, based on a couple of analytically solvable 1D-Ising chains. The presented novel approach is based on rigorous statistical-thermodynamic principles and involves modeling the free energy contribution of an added inter-chain bond $\Delta A_{\mathrm{bond}}(\beta ,N)$ as function of inverse temperature $\beta$ and lattice size N. The identified simple analytic expression for $\Delta A_{\mathrm{bond}}$ is fitted to exact results of a series of finite-size quadratic $N\times N$-systems and enables straightforward and instantaneous calculation of thermodynamic quantities of interest, such as free energy and heat capacity for systems of an arbitrary size. This approach is not only interesting from a fundamental perspective with respect to the possible transfer to a 3D-Ising model, but also from an application-driven viewpoint in the context of (Li-ion) batteries where it could be applied to describe intercalation mechanisms. Full article

The coherent energy transfer between two identical two-level systems is investigated. Here, the first quantum system plays the role of a charger, while the second can be seen as a quantum battery. Firstly, a direct energy transfer between the two objects is considered and then compared to a transfer mediated by an additional intermediate two-level system. In this latter case, it is possible to distinguish between a two-step process, where the energy is firstly transferred from the charger to the mediator and only after from the mediator to the battery, and a single-step in which the two transfers occurs simultaneously. The differences between these configurations are discussed in the framework of an analytically solvable model completing what recently discussed in literature. Full article

Simple interpolation formulas are proposed for the description of the renormalization group (RG) scale dependences of the gravitational couplings in the framework of the 2-parameters Einstein-Hilbert (EH) theory of gravity and applied to a simple, analytically solvable, spatially homogeneous and isotropic, spatially flat model universe. The analytical solution is found in two schemes incorporating different methods of the determination of the conversion rule $k\left(t\right)$ of the RG scale k to the cosmological time t. In the case of the discussed model these schemes turn out to yield identical cosmological evolution. Explicit analytical formulas are found for the conversion rule $k\left(t\right)$ as well as for the characteristic time scales $t_G$ and $t_{\mathsf{\Lambda }}>t_G$ corresponding to the dynamical energy scales $k_G$ and $k_{\mathsf{\Lambda }}$, respectively, arising form the RG analysis of the EH theory. It is shown that there exists a model-dependent time scale $t_d$ ($t_G\le t_d<t_{\mathsf{\Lambda }}$) at which the accelerating expansion changes to the decelerating one. It is shown that the evolution runs from a well-identified cosmological fixed point to another one. As a by-product we show that the entropy of the system decreases monotonically in the interval $0<t\le t_{\mathsf{\Lambda }}$ due to the quantum effects. Full article

A spin-boson-like model with two interacting qubits is analysed. The model turns out to be exactly solvable since it is characterized by the exchange symmetry between the two spins. The explicit expressions of eigenstates and eigenenergies make it possible to analytically unveil the occurrence of first-order quantum phase transitions. The latter are physically relevant since they are characterized by abrupt changes in the two-spin subsystem concurrence, in the net spin magnetization and in the mean photon number. Full article

This research presents a nonlinear adaptive fuzzy control method as an analytical design and a simple control structure for the trajectory tracking problem in wheeled mobile robots with skew symmetrical property. For this trajectory tracking problem in wheeled mobile robots, it is not easy to find an analytical adaptive fuzzy control solution due to the complicated error dynamics between the controlled wheeled mobile robots and desired trajectories. For deriving the analytical adaptive fuzzy control law of this trajectory tracking problem, a filter link is firstly adopted to find the solvable error dynamics, then the research is based on the skew symmetrical property of the transformed error dynamics. This proposed nonlinear adaptive fuzzy control solution has the advantages of low computational resource consumption and elimination of modeling uncertainties. From the results for tracking two simulation scenarios (an S type trajectory and a square trajectory), the proposed nonlinear adaptive fuzzy control method demonstrates a satisfactory trajectory tracking performance for the trajectory tracking problem in wheeled mobile robots with huge modeling uncertainties and outperforms the existing H₂ control method. Full article

The Cahn–Hilliard–Navier–Stokes model is extensively used for simulating two-phase incompressible fluid flows. With the absence of exterior force, this model satisfies the energy dissipation law. The present work focuses on developing a linear, decoupled, and energy dissipation-preserving time-marching scheme for the hydrodynamics coupled Cahn–Hilliard model. An efficient time-dependent auxiliary variable approach is first introduced to design equivalent equations. Based on equivalent forms, a BDF2-type linear scheme is constructed. In each time step, the unique solvability and the energy dissipation law can be analytically estimated. To enhance the energy stability and the consistency, we correct the modified energy by a practical relaxation technique. Using the finite difference method in space, the fully discrete scheme is described, and the numerical solutions can be separately implemented. Numerical results indicate that the proposed scheme has desired accuracy, consistency, and energy stability. Moreover, the flow-coupled phase separation, the falling droplet, and the dripping droplet are well simulated. Full article