Closed-form Solution Research Articles

Closed-Form Solutions of Injection-Driven Flow and Heat Transfer Inside an Inclined Horizontal Filter Chamber

Abstract The closed-form solutions of models provide operators and designers with a better understanding of how the systems perform practically, thus improving critical industrial production operations. Due to this importance, the case study at hand seeks to find closed-form solutions of the internal momentum and temperature variation during the filtration process to advance fluid purification. Lie group analysis is used to transform a system of equations representing the flow and heat transfer into a solvable system without changing the dynamics of the case study. The transformed solvable system is then integrated to find closed-form solutions of the internal velocity (momentum) and temperature variation. The obtained closed-form solutions are then used to analyze the effects of physical parameters arising from the process dynamics to find combinations of parameters that yield maximum permeates outflow. The analysis conveys that the internal fluid velocity increases when enhancing the permeation parameter and minimizing Reynolds number, wave speed parameter, and chamber height.

Closed-form analytical solution for the transfer matrix based on Pendry-MacKinnon discrete Maxwell's equations

Closed-form analytical solution for the transfer matrix based on Pendry-MacKinnon discrete Maxwell's equations

The Transitional Dynamics of Debt: Productivity Shocks, Fiscal Policy and Economic Growth

We propose a two-sector endogenous growth model with both public and private production. The government sector produces an output that enters both the utility function and the production function, financed by both taxes and borrowing, such that public expenditure and public finance are independently determined. We impose a homogeneous stylized labour productivity supply shock to country specific simulations, coinciding with hypothetical year 2020, and consider their respective heterogeneous fiscal responses. The simulations depict the transitional dynamics in both developed and less developed economies to the simultaneous labour and policy shocks. Using advanced software applications, explicitly suited to model continuous time mathematics, we simulate closed form solutions to continuous time general equilibria that are defined by a three-dimensional modified golden rule across consumption, capital and debt. The model represents a baseline for analysis of the short to medium term impacts of shocks, both exogenous or policy induced, within a richly defined dynamic general equilibrium. Our methodology extends the ability to analyze transitional dynamics in multidimensional models where hitherto analysis has been confined to the long run equilibria.

Optimal Consumption and Investment with Income Adjustment and Borrowing Constraints

In this paper, we address the utility maximization problem of an infinitely lived agent who has the option to increase their income. The agent can increase their income at any time, but doing so incurs a wealth cost proportional to the amount of the increase. To prevent the agent from infinitely increasing their income and borrowing against future income, we additionally consider a non-negative wealth constraint that prohibits borrowing based on future income. This utility maximization problem is a mixture of stochastic control, where the agent chooses consumption and investment, and singular control, where the agent chooses a non-decreasing income process. To solve this non-trivial and challenging problem, we derive the Hamilton–Jacobi–Bellman (HJB) equation with a gradient constraint using the dynamic programming principle (DPP). Then, using the guess-and-verify method and a linearization technique, we obtain a closed-form solution to the HJB equation and, based on this, find the optimal strategy.

Closed-Form Solution for Scale-Dependent Frequencies of Shear Deformable Cellular Microplates Coated by Piezoelectric Polymeric Skins Consisting of FG Distributed CNTs

In this work, a rectangular cellular microplate is taken into consideration which is embedded between two functionally graded carbon nanotube-reinforced composite layers that have the piezoelectric ability. Different patterns for the carbon nanotube dispersion are considered. Moreover, an external electrical voltage is applied to them. The displacements are described based on a higher-order shear deformation theory which is called hyperbolic theory and the size influence is captured via the modified couple stress formulations. The governing motion equations are then derived using the variational technique and Hamilton’s principle. Then, for simply supported edge conditions, a closed-form solution is provided. Next, it turns to validate the results in simpler states, and after that, the effect of the most prominent parameters on the results is investigated. It is observed that by increasing the external applied voltage to the face sheets, the frequencies are reduced. Also, the natural frequencies have a tendency to decrease as the dimensionless material length scale parameter increases. This study’s outcomes may help design and manufacture micro-/nano-electro systems, sensors and actuators, small-scale devices, and other engineering structures.

Optimal approximate minimization of one-letter weighted finite automata

Abstract In this paper, we study the approximate minimization problem of weighted finite automata (WFAs): to compute the best possible approximation of a WFA given a bound on the number of states. By reformulating the problem in terms of Hankel matrices, we leverage classical results on the approximation of Hankel operators, namely the celebrated Adamyan-Arov-Krein (AAK) theory. We solve the optimal spectral-norm approximate minimization problem for irredundant WFAs with real weights, defined over a one-letter alphabet. We present a theoretical analysis based on AAK theory and bounds on the quality of the approximation in the spectral norm and $\ell ^2$ norm. Moreover, we provide a closed-form solution, and an algorithm, to compute the optimal approximation of a given size in polynomial time.

Exact Solution of Nonlinear TVMD Based on Maximizing Damping Enhancement Effect for Acceleration Response Control

The tuned viscous mass damper (TVMD) exhibits superior control effect and energy dissipation capability compared to the viscous damper (VD). Existing literature has primarily focused on the control of absolute acceleration response in structures by linear TVMD, with limited research on nonlinear damper for controlling structural performance and efficiency. This study proposes a methodology for determining the optimal parameters of linear TVMD. The closed-form solution of the nonlinear TVMD is derived through stochastic linearization based on the linear TVMD, aiming to minimize the damping ratio of TVMD by utilizing the damping enhancement effect of the system. For lightly damped structures, the closed-form solution remains a reliable method for accurately approximating TVMD design parameters. Compared to traditional fixed-point theory, the TVMD developed through the proposed methodology demonstrates superior control performance, especially in terms of absolute acceleration, with a smaller damping ratio and minimal deviation from the optimal state of the TVMD. Additionally, the study investigated the influence of the damping index on the optimal parameters and the efficacy of the TVMD in vibration control. It is observed that reducing the damping index can improve structural performance, particularly in long-period structures with small mass ratios, although it may reduce the efficiency of the TVMD. Notably, TVMDs with lower mass ratios maintain higher efficiency regardless of the structural natural period. The damping index does not affect the optimal frequency ratio or the mean square response. Ultimately, the effectiveness of the optimization method is displayed through the examination of response spectra following 22 real seismic events, showcasing its adaptability across various structural types. The comparison of the root-mean-square response between the VD and TVMD across different periods shows consistent results, with a maximum deviation of only 10%. The TVMD is highly effective at regulating the structural response, excelling particularly in controlling the absolute acceleration of long-period structures, where it reduces maximum absolute acceleration by 20% more than displacement. The proposed exact solution rapidly and precisely identifies the design parameters of nonlinear TVMD by the desired specifications, thereby offering theoretical guidance for practical engineering applications.

Rapid Assessment Method of Buckling Stability of Bridge Pier–Cap–Pile System Under Scour Effect

A novel analytical method for evaluating the buckling stability of the pier–cap–pile system under scour effect is proposed, and furthermore, the concept of rapid assessment of bridge buckling stability based on natural frequency is innovatively introduced in this paper. First, the total potential energy function of the bridge pier–cap–pile system is established, and the closed-form solution of structural system critical buckling load under scouring is solved according to the principle of stationary potential energy. Second, the sensitivity analyses of pier height, pier width, bending stiffness reduction in pier, pile slenderness ratio, pile bending stiffness reduction, m-value of soil and structural gravity to the buckling load are carried out. On this basis, 60 sets of sensitive parameter sample combinations are extracted by the Latin Hypercube Sampling algorithm, and a proxy model relating the critical buckling load to scour depth and the sensitive parameters are then developed using the McQuarter global optimization algorithm. Finally, the regression relationship model between the critical buckling load and the natural frequency is formed by combining the proxy model and the existing relationship between the scour depth and the first-order natural frequency. The feasibility of the proposed assessment method is demonstrated by the finite element method using a bridge case study, which provides an insight and a new means of quantitatively evaluating the safety of bridges under scour effect.

Robustness meets co-jumps: optimal consumption and portfolio choice with derivatives

In this paper, we study a robust, dynamic, continuous-time optimal consumption and portfolio allocation problem for investors with recursive preferences who have access to both stock and derivatives markets. We assume the stock price process follows a stochastic volatility model, with instantaneous precision as the unique state variable, allowing for discontinuities in all the dynamics. We obtain a closed-form approximate solution up to a system of ODEs to the optimization problem for a non-unitary value of the elasticity of intertemporal substitution of consumption, being able to derive an exact solution as a particular case. Our theoretical findings show that the optimal policies are remarkably affected by the ambiguity-aversion parameters to diffusive and jump risks. A detailed numerical analysis confirms the effectiveness of our theoretical results on real data. Finally, we prove that investors who do not believe in ambiguity may suffer considerable wealth losses.

Fracture Toughness of Ordinary Plain Concrete Under Three-Point Bending Based on Double-K and Boundary Effect Fracture Models.

Fracture tests are a necessary means to obtain the fracture properties of concrete, which are crucial material parameters for the fracture analysis of concrete structures. This study aims to fill the gap of insufficient test results on the fracture toughness of widely used ordinary C40~C60 concrete. A three-point bending fracture test was conducted on 28 plain concrete and 6 reinforced concrete single-edge notched beam specimens with various depths of prefabricated notches. The results are reported, including the failure pattern, crack initiation load, peak load, and complete load versus crack mouth opening displacement curves. The cracking load showed significant variation due to differences in notch prefabrication and aggregate distribution, while the peak load decreased nonlinearly with an increase in the notch-to-height ratio. The reinforced concrete beams showed a significantly higher peak load than the plain concrete beams, attributed to the restraint of steel reinforcement, but the measured cracking load was comparable. A compliance versus notch-to-height ratio curve was derived for future applications, such as estimating crack length in crack growth rate tests. Finally, fracture toughness was determined based on the double-K fracture model and the boundary effect model. The average fracture toughness value for C50 concrete from this study was 2.0 MPa·m, slightly smaller than that of lower-strength concrete, indicating the strength and ductility dependency of concrete fracture toughness. The fracture toughness calculated from the two models is consistent, and both methods employ a closed-form solution and are practical to use. The derived fracture toughness was insensitive to the discrete parameters in the boundary effect model. The insights gained from this study significantly contribute to our understanding of the fracture toughness properties of ordinary structural concrete, highlighting its potential to shape future studies and applications in the field.

Simulation model calibration with dynamic stratification and adaptive sampling

ABSTRACT Calibrating simulation models that take large quantities of multi-dimensional data as input is a hard simulation optimization problem. Existing adaptive sampling strategies offer a methodological solution. However, they may not sufficiently reduce the computational cost for estimation and solution algorithm’s progress within a limited budget due to extreme noise levels and heteroscedasticity of system responses. We propose integrating stratification with adaptive sampling for the purpose of efficiency in optimization. Stratification can exploit local dependence in the simulation inputs and outputs. Yet, the state-of-the-art does not provide a full capability to adaptively stratify the data as different solution alternatives are evaluated. We devise two procedures for data-driven calibration problems that involve a large dataset with multiple covariates to calibrate models within a fixed overall simulation budget. The first approach dynamically stratifies the input data using binary trees, while the second approach uses closed-form solutions based on linearity assumptions between the objective function and concomitant variables. We find that dynamical adjustment of stratification structure accelerates optimization and reduces run-to-run variability in generated solutions. Our case study for calibrating a wind power simulation model, widely used in the wind industry, using the proposed stratified adaptive sampling, shows better-calibrated parameters under a limited budget.

Trajectory Generation for Legged Robots Based on a Closed-Form Solution of Centroidal Dynamics

Trajectory Generation for Legged Robots Based on a Closed-Form Solution of Centroidal Dynamics

Analytical assessment of suspension bridge's 3D curved cable configuration and cable clamp pre-installation angle considering the main cable torsional and flexural stiffnesses

For a suspension bridge with a spatial cable system, the 3D curved main cable undergoes large lateral and torsional deformations during construction, which increases the difficulty of construction control. If using the traditional ideal flexible cable assumption, the torsional deformation cannot be analyzed. Therefore, incorporating the main cable's torsional and flexural stiffnesses in shape-finding analysis remains highly challenging. This study develops an analytical method for determining the target configuration of the main cable by applying the multi-segment catenary method and the Cosserat rod model. The closed-form solution of the geometrically exact force-displacement-strain relationships is derived, comprehensively considering the tension, shear, bending, and torsion of the main cable, as well as the initial curvatures and axial strains in the reference configuration. Then a two-layer shape-finding framework is established, with the first layer being recursive calculations of the cable shape and the second layer being numerical calculations of the nonlinear governing equations using the Levenberg-Marquardt method. Furthermore, a novel method for calculating the pre-deflection angle of the cable clamp in the free cable state is presented for the first time. A classic cantilever model and a suspension bridge with the spatial main cable are studied to investigate the accuracy of the proposed algorithms. Numerical results indicate that when the spatial main cable is twisted, the bottom edge of the cable cross-section moves outward along the transverse direction of the bridge. The torsion of the main cable includes both elastic deformation and rigid body displacement caused by the bidirectional bending effect. At the mid-span, the torsional angle of the main cable cross-section is 10.088°, and the pre-deflection angle of the cable clamp at the mid-span should be set to 10.851°. Moreover, the target configuration is highly sensitive to the flexural and torsional stiffnesses of the main cable while the effect of shear deformations on cable configurations can be ignored.

FairDRO: Group fairness regularization via classwise robust optimization

Existing group fairness-aware training methods fall into two categories: re-weighting underrepresented groups according to certain rules, or using regularization terms such as smoothed approximations of fairness metrics or surrogate statistical quantities. While each category has its own strength in applicability or performance when compared to each other, their successful performances are typically limited to specific cases. To that end, we propose a new approach called FairDRO, which takes advantage of both categories through a classwise group distributionally robust optimization (DRO) framework. Our method unifies re-weighting and regularization by incorporating a well-justified group fairness metric into the objective as regularization, but solving it through a principled re-weighting strategy. To optimize our resulting objective efficiently, we adopt an iterative algorithm and consequently develop two variants of FairDRO algorithm depending on the choice of surrogate loss. For in-depth understanding, we derive three theoretical results: (i) a closed-form solution for the correct re-weights; (ii) justifications for using the surrogate losses; and (iii) a convergence analysis of our method. Experimental results show that our algorithms consistently achieve state-of-the-art performance in accuracy-fairness trade-offs across multiple benchmarks, demonstrating scalability and broad applicability compared to existing methods.

Hygrothermoelastic analysis of the nano-circular plate with memory effect

In hygrothermal environments, the coupling effects of temperature and moisture substantially impact deflection and stresses play a significant role. This study presents a coupled hygrothermoelastic model with non-Fourier and non-Fick effects established by introducing relaxation times or phase lags of heat and moisture flux accompanied by memory-dependent derivatives. The boundary value problem is formulated by considering a thin circular plate as an exemplary example, where the perimetric edge is clamped. The upper and lower edges of the plate is subjected to zero temperature, whereas the curved surface is exposed to hygrothermal shock. The closed-form solution of temperature and moisture distribution is obtained via the integral transform approach. The Fourier series expansion approach is used to calculate the numerical Laplace inversion. The effects of both heat and moisture flux relaxation times on the thermal deflection/stresses of the plate are analyzed and illustrated graphically.

Overview of Tensor-Based Cooperative MIMO Communication Systems—Part 2: Semi-Blind Receivers

Cooperative MIMO communication systems play an important role in the development of future sixth-generation (6G) wireless systems incorporating new technologies such as massive MIMO relay systems, dual-polarized antenna arrays, millimeter-wave communications, and, more recently, communications assisted using intelligent reflecting surfaces (IRSs), and unmanned aerial vehicles (UAVs). In a companion paper, we provided an overview of cooperative communication systems from a tensor modeling perspective. The objective of the present paper is to provide a comprehensive tutorial on semi-blind receivers for MIMO one-way two-hop relay systems, allowing the joint estimation of transmitted symbols and individual communication channels with only a few pilot symbols. After a reminder of some tensor prerequisites, we present an overview of tensor models, with a detailed, unified, and original description of two classes of tensor decomposition frequently used in the design of relay systems, namely nested CPD/PARAFAC and nested Tucker decomposition (TD). Some new variants of nested models are introduced. Uniqueness and identifiability conditions, depending on the algorithm used to estimate the parameters of these models, are established. Two families of algorithms are presented: iterative algorithms based on alternating least squares (ALS) and closed-form solutions using Khatri–Rao and Kronecker factorization methods, which consist of SVD-based rank-one matrix or tensor approximations. In a second part of the paper, the overview of cooperative communication systems is completed before presenting several two-hop relay systems using different codings and configurations in terms of relaying protocol (AF/DF) and channel modeling. The aim of this presentation is firstly to show how these choices lead to different nested tensor models for the signals received at destination. Then, by capitalizing on these models and their correspondence with the generic models studied in the first part, we derive semi-blind receivers to jointly estimate the transmitted symbols and the individual communication channels for each relay system considered. In a third part, extensive Monte Carlo simulation results are presented to compare the performance of relay systems and associated semi-blind receivers in terms of the symbol error rate (SER) and channel estimate normalized mean-square error (NMSE). Their computation time is also compared. Finally, some perspectives are drawn for future research work.

Optimal Asset Allocation of Pension Funds with a Portfolio Constraint Change

We investigated the asset allocation problem of a pension fund in which the constraints on the risky investment proportion change owing to local regulations or the fund’s policy changes. We assumed that the fund manager has a utility function with respect to the funding ratio of the pension fund. A dynamic programming approach was employed and we obtained a closed-form solution to the problem. To numerically measure the effect of such changes on asset allocation, we define the concept of policy change equivalent return (PCER) as the additional expected return for the value function of the case without such changes at the same level as the value function of the case with them. In the numerical analysis, we analyzed both the effect of regime switching on market parameters and policy change on the PCER and showed the effectiveness of introducing policy changes to asset allocation.

Modeling and Performance Evaluation of a Cellular Network with OMA and NOMA Users with Batch Arrivals by Means of an M[X]/M/S/0 Model

Nowadays, efficient spectrum usage is one of the most important design principles to take into account in wireless communications due to the exponential growth of mobile devices. In that sense, solutions such as Non-Orthogonal Multiple Access (NOMA) and cognitive radio (CR) have been proposed. In essence, NOMA allows some interference level by using non-orthogonal resource allocation with a tolerable increase in receiver complexity employing successive interference cancellation (SIC). In this work, a novel mathematical model of teletraffic for users performing accessment, simultaneously, by means of Orthogonal Multiple Access (OMA) and NOMA, is developed using a Markovian process that considers bursts of arrivals to model the access schemes. This novel procedure implies a closed-form solution of the proposed system compared to other works where these parameters are estimated assuming the moment generating function obtained with approximation models. The model is validated with a discrete event simulator, considering different scenarios and simulation conditions. The simulation results are in agreement with the mathematical solution proposed.

The simple analytics of the environmental Kuznets curve: a reformulation

PurposeThis paper aims to review the relationship between per capita income and aggregate emission in an economy populated by rational agents. According to the environmental Kuznets curve (EKC) hypothesis, pollution–income relationship is an inverted U-shaped curve. This paper aims to derive that relationship in an endogenous way and extends the relevant literature in an important way.Design/methodology/approachThis paper formulated a general equilibrium model of homogenous population with identical tastes and preferences. Production side is modelled with firms operating in a monopolistically competitive environment. The approach is modelling the economy in an analytical way so that closed form solutions can be achieved. Model simulations have also been performed to get a clear view of results.FindingsThis study shows that increasing returns to scale in abatement technology ceases to be a sufficient condition for the generation of the EKC hypothesis. The general equilibrium structure of the model allows studying the endogenous evolution of income, emissions and prices of the abatement goods in a unified setting.Originality/valueThe paper is novel and original in nature. The results are new in the literature. These results extend and generalise the previous research work in this area in an important way. The sufficient condition that is obtained in this study limits the applicability of EKC in an otherwise identical economy as used in previous literature. Therefore, this paper adds value to the ongoing research related to EKC.

Closed-form solutions for Bernoulli and compound Poisson branching processes in random environments*

Abstract For branching processes, the generating functions for the limit distributions of the so-called ratios of probabilities of rare events satisfy Schröder-type integral–functional equations. With the exception of a few special cases, the corresponding equations cannot be solved analytically. I found a large class of Poisson-type offspring distributions for which the Schröder-type functional equations can be solved analytically. Moreover, for the asymptotics of limit distributions, the power and constant factors can be written explicitly. As a bonus, Bernoulli branching processes in random environments are treated. The beauty of this example is that the explicit formula for the generating function is unknown. Nevertheless, the closed-form expressions for the power and constant factors in the asymptotic can be written with the help of some effective techniques. The asymptotic expansion also contains oscillatory terms absent in the Poisson case. It is shown that at power three in the Bernoulli binomial kernels, short-phase discrete oscillations turn into long-phase continuous ones.