Closed-form Solution Research Articles

Vaccination for communicable endemic diseases: optimal allocation of initial and booster vaccine doses.

For some communicable endemic diseases (e.g., influenza, COVID-19), vaccination is an effective means of preventing the spread of infection and reducing mortality, but must be augmented over time with vaccine booster doses. We consider the problem of optimally allocating a limited supply of vaccines over time between different subgroups of a population and between initial versus booster vaccine doses, allowing for multiple booster doses. We first consider an SIS model with interacting population groups and four different objectives: those of minimizing cumulative infections, deaths, life years lost, or quality-adjusted life years lost due to death. We solve the problem sequentially: for each time period, we approximate the system dynamics using Taylor series expansions, and reduce the problem to a piecewise linear convex optimization problem for which we derive intuitive closed-form solutions. We then extend the analysis to the case of an SEIS model. In both cases vaccines are allocated to groups based on their priority order until the vaccine supply is exhausted. Numerical simulations show that our analytical solutions achieve results that are close to optimal with objective function values significantly better than would be obtained using simple allocation rules such as allocation proportional to population group size. In addition to being accurate and interpretable, the solutions are easy to implement in practice. Interpretable models are particularly important in public health decision making.

A closed-form solution of dowel action based on beam on elastic foundation theory and fracture mechanics

This paper proposes a new mechanical model to describe the dowel action with the aim of using the model to gain a deeper understanding of the unstable dowel splitting cracking observed in shear experiments of beams without shear reinforcement. The model was developed by combining beam on elastic foundation (BEF) theory and fracture mechanics. The proposed model is able to predict the whole evolution process of dowel action until the propagation of the dowel splitting crack becomes unstable. The model theoretically proves that the development of a dowel splitting crack can become unstable under certain conditions, therefore leading to the unstable shear failure of the whole member. In addition to the derivation of the analytical model, the paper also validates the model using data from the literature. Finally, an analytical solution of the critical shear displacement that triggers the unstable dowel splitting crack is derived. It can be used to improve the failure criterion initially proposed in the Critical Shear Displacement Theory (CSDT).

Flexural performance of GFRP-sheathed cold-formed steel composite panel filled with lightweight phosphogypsum

To develop lightweight, high-strength and environmentally friendly prefabricated panels, this paper proposed a novel GFRP plate-sheathed cold-formed steel (CFS) composite panel (GFRP-CFS panel) filled with lightweight phosphogypsum (LPG) and investigated its flexural performance experimentally, numerically and theoretically. Firstly, nine full-scale specimens were subjected to four-point bending test. The effects of the number, web height, thickness and arrangement of the longitudinal CFS studs, the number of lateral CFS studs, and the support connection on the flexural performance were discussed. Results indicate that the failure mode of the GFRP-CFS panels was debonding of GFRP plate in the compression zone at midspan and local buckling of CFS below the loading point. The strain of GFRP plate always lagged behind that of the CFS stud at the corresponding position, and the built-up cross section of the panel no longer conformed to the plane-section assumption. In addition, a nonlinear finite element modelling (FEM) analysis based on DIANA software was conducted to validate the test results. The load-deflection response, strain response and failure mode from the FEM analyses were in good agreement with the test results. Furthermore, based on the theoretical analysis considering the bond-slip effect between GFRP and CFS, a closed-form solution for the GFRP utilization coefficient was derived. The effect of different variables on the GFRP utilization coefficient was analyzed through parametric analysis. Finally, the calculation method for the flexural strength and ultimate deflection of GFRP-CFS panels were established, and the predicted results were in good agreement with test results. Therefore, the developed GFRP-CFS panels have excellent potential for prefabricated floor and wall applications.

Neural network approach to portfolio optimization with leverage constraints: a case study on high inflation investment

Motivated by the current global high inflation scenario, we aim to discover a dynamic multi-period allocation strategy to optimally outperform a passive benchmark while adhering to a bounded leverage limit. We formulate an optimal control problem to outperform a benchmark portfolio throughout the investment horizon. To obtain strategies under the bounded leverage constraint among other realistic constraints, we propose a novel leverage-feasible neural network (LFNN) to represent the control, which converts the original constrained optimization problem into an unconstrained optimization problem that is computationally feasible with gradient descent, without dynamic programming. We establish mathematically that the LFNN approximation can yield a solution that is arbitrarily close to the solution of the original optimal control problem with bounded leverage. We further validate the performance of the LFNN empirically by deriving a closed-form solution under jump-diffusion asset price models and show that a shallow LFNN model achieves comparable results on synthetic data. In the case study, we apply the LFNN approach to a four-asset investment scenario with bootstrap-resampled asset returns from the filtered high inflation regimes. The LFNN strategy is shown to consistently outperform the passive benchmark strategy by about 200 bps (median annualized return), with a greater than 90% probability of outperforming the benchmark at the end of the investment horizon.

Discriminative Subspace Learning With Adaptive Graph Regularization

Abstract Many subspace learning methods based on low-rank representation employ the nearest neighborhood graph to preserve the local structure. However, in these methods, the nearest neighborhood graph is a binary matrix, which fails to precisely capture the similarity between distinct samples. Additionally, these methods need to manually select an appropriate number of neighbors, and they cannot adaptively update the similarity graph during projection learning. To tackle these issues, we introduce Discriminative Subspace Learning with Adaptive Graph Regularization (DSL_AGR), an innovative unsupervised subspace learning method that integrates low-rank representation, adaptive graph learning and nonnegative representation into a framework. DSL_AGR introduces a low-rank constraint to capture the global structure of the data and extract more discriminative information. Furthermore, a novel graph regularization term in DSL_AGR is guided by nonnegative representations to enhance the capability of capturing the local structure. Since closed-form solutions for the proposed method are not easily obtained, we devise an iterative optimization algorithm for its resolution. We also analyze the computational complexity and convergence of DSL_AGR. Extensive experiments on real-world datasets demonstrate that the proposed method achieves competitive performance compared with other state-of-the-art methods.

Exact solutions for the elastic-plastic response of functionally graded pipe under external pressure

This study investigates the influence of material gradient on the elastoplastic response of functionally graded (FG) thick-walled pipes subjected to external pressure. Closed-form solutions are derived for radial and circumferential stresses across various stages: purely elastic, partially plastic, and post-unloading from plastic regimes. Both the elastic modulus and yield stress exhibit radial variation defined by power-law functions, while Poisson's ratio remains constant. The stress distribution within the pipe depends not only on the external pressure but also on the material gradient index and the geometric dimensions of the pipe. A method is presented to accurately determine the first yielding position in a thin-walled tube based on the material gradient index by employing Tresca's yield criterion. Plastic zones nucleate at one or both surfaces as external pressure increases. The propagation of elastoplastic interfaces and stress distributions within each zone are examined in detail. Notably, following plastic deformation and complete unloading, FG thick-walled pipes may exhibit re-emergence of plastic deformation, depending on the pre-unloading stress state and the gradient of the elastic modulus. The conclusions of this work expected to aid in the design of FG pressure vessels and improve their load-carrying capacity and safety.

An analytical study on the influence of magnetic field-dependent viscosity on viscous and ohmic dissipative second-grade fluid boundary layers

This article aims to investigate the magnetohydrodynamic (MHD) boundary layer flow of a second-grade non-Newtonian fluid over a horizontally stretching sheet, considering magnetic field-dependent (MFD) viscosity, as well as viscous and Ohmic dissipations. Analytical solutions for previously unexplored momentum and heat transfer equations involving MFD viscosity are derived. Two boundary conditions, namely Prescribed Surface Temperature (PST) and Prescribed Heat Flux (PHF), are taken into account. Governing dimensional partial differential equations (PDEs) are transformed into non-dimensional ordinary differential equations (ODEs) using similarity transformations. Closed-form analytical solutions for flow are derived, considering stretching velocity, MFD viscosity, second-grade fluid properties, and suction impacts. Heat transfer equations are transformed into Gauss-hypergeometric form, yielding solutions in terms of confluent hypergeometric functions. Analytical expressions for skin friction, local Nusselt number, and non-dimensional wall temperature are derived. A unique solution is obtained for the flow equation. It is found that both second-grade and magnetic viscosity parameters expand the momentum boundary layer while the magnetic parameter reduces thickness. Thermal boundary layer thickness increases with a higher second-grade parameter, magnetic viscosity, and Eckert number. Moreover, analytical solutions are validated against published results for a special case.

Multiscale reweighted smoothing regularization in curvelet domain for hyperspectral image denoising

ABSTRACT Hyperspectral images are easily contaminated by multi-modal noise in the process of data acquisition. Generally, hyperspectral signatures and noise appear unexpectedly scattered in the spatial-spectral domain. Conversely, in the transform domain, the principle components of an image are gathered in the low-frequency bands, while the details and noise often occur in high-frequency parts. The existing denoising methods usually divide the whole frequency domain into two parts, that is, high frequency and low frequency, which can not make the most of the knowledge contained at different frequencies or scales, especially in the multiscale transform domain. Furthermore, aiming to restore a clean image from the noise-corrupted data, hyperspectral image denoising often amounts to an ill-posed problem, in which reshaping cubic data into matrix form results in the loss of structure information. To address the aforementioned issues, this paper incorporates the reweighted smoothing regularization in the transform domain into the noise-perturbed degradation model to reformulate a novel optimization problem. First, the multiscale curvelet is leveraged to transform the spatial matrix into frequency domain, where the coefficients are carefully designed to weight the components at different scales and orientations. Second, l 1 -norm is imposed on the abundance term to promote sparsity in the transform domain for performance enhancement. Finally, using proximal alternating optimization, an efficient algorithm is well developed to obtain the closed-form solutions to the proposed denoising problem. The experimental results on several synthetic and real datasets, illustrate that the proposed method improves PSNR by at least 1.5 dB and reduces ERGAS error by more than 35% than the state-of-the-art approaches under the multi-modal noise mixed environments, which verifies the validity of regularization terms in hyperspectral images denoising.

A Note on Exact Closed-Form Solutions for Lamb’s Problem with an Arbitrary Poisson’s Ratio

Abstract Lamb’s Problem, a cornerstone in theoretical seismology, investigates the displacement of an elastic half-space due to surficial or buried point sources. Although the exact closed-form solution for surface displacements from surficial sources is well established, the solutions for buried sources in the 3D case have remained limited, either confined to integral forms or applicable only for certain Poisson’s ratios. This study addresses this gap by generalizing the existing closed-form solutions to accommodate Poisson’s ratio for buried sources. Our solutions are expressed in terms of elementary algebraic expressions, elliptic integrals, and Gauss hypergeometric functions. The derived solution is validated by numerical tests. With the newly derived solution, we also explore the properties of the leaking mode, which stems from the complex conjugate roots of the Rayleigh equation.

A Power Method for Computing the Dominant Eigenvalue of a Dual Quaternion Hermitian Matrix

In this paper, we first study the projections onto the set of unit dual quaternions, and the set of dual quaternion vectors with unit norms. Then we propose a power method for computing the dominant eigenvalue of a dual quaternion Hermitian matrix. For a strict dominant eigenvalue, we show the sequence generated by the power method converges to the dominant eigenvalue and its corresponding eigenvector linearly. For a general dominant eigenvalue, we establish linear convergence of the standard part of the dominant eigenvalue. Based upon these, we reformulate the simultaneous localization and mapping problem as a rank-one dual quaternion completion problem. A two-block coordinate descent method is proposed to solve this problem. One block has a closed-form solution and the other block is the best rank-one approximation problem of a dual quaternion Hermitian matrix, which can be computed by the power method. Numerical experiments are presented to show the efficiency of our proposed power method.

Lie symmetry analysis and solitary wave solution of biofilm model Allen-Cahn

The investigation presented in this study delves into the analysis of Lie symmetries for the bistable Allen-Cahn (BAC) equation with a quartic potential, specifically applied to the biofilm model. By employing the Lie symmetry method, we have acquired the Lie infinitesimal generators for the considered model. Using a transformation method, the nonlinear partial differential equations (NPDEs) are converted into various nonlinear ordinary differential equations (NLODEs), providing the numerous closed-form solitary wave solutions. The obtained solutions manifest in various forms including dark, bright, kink, anti-kink, and periodic types using diverse strategies. To enhance the physical interpretation, the study presents 3D, 2D, and contour plots of the acquired solutions. Every graph’s wave-like structure contains information about the structural behaviour of the bacteria that build biofilms on surfaces where rectangles have different densities. This analysis enhances comprehension of the complex dynamics present in areas like fluid dynamics, fiber optics, biology, ocean physics, coastal engineering, and nonlinear complex physical systems.

Error performance and capacity of subcarrier BPSK modulated THz wireless systems with pointing errors

Terahertz (THz) technology is poised to play a pivotal role in the future sixth-generation (6 G) wireless systems, offering substantial benefits with its extremely high bandwidth and rapid transmission speeds. This paper explores subcarrier-modulated THz wireless systems, analyzing their performance under the log-normal and gamma-gamma models. It provides a detailed examination of how composite turbulence and pointing errors impact the system performance. Closed-form solutions for the bit error rate and outage probability are derived for various conditions of turbulence and pointing errors. Furthermore, the paper delves into the asymptotic error performance, presenting numerical results that confirm the accuracy of the derived solutions across different propagation distances, turbulence strengths, and pointing error scenarios. The study offers critical insights into how different conditions affect the system performance, which is essential for characterizing subcarrier THz links in forthcoming 6 G deployments.

Exploring the relationship between the International Roughness Index (IRI) and the ISO 8608 roughness coefficient (Gd (n 0))

ABSTRACT Globally, diverse agencies employ varied metrics to gauge road roughness, complicating the exchange of information for pavement maintenance. This study addresses this challenge by establishing a theoretical connection between the International Roughness Index (IRI) and the ISO 8608 Power Spectral Density (PSD)-based roughness coefficient, namely Gd (n 0). Based on random vibration theory, this paper first deduces a closed-form explicit solution for IRI related to Gd (n 0), solidifying a theoretical link. Next, the variability and convergence of the proposed solution are examined through Monte Carlo simulations using a quarter-car vehicle model with the Golden Car parameters. Then, we compare computed results from this study with those existing solutions from past studies, verifying that the proposed solution captures the curve trend well and agrees with the average value of existing solutions based on field measurement data. Finally, this paper proposes the IRI values corresponding to various ISO road class thresholds and a new Gd (n 0) scale for ride quality assessment of highways.

High-resolution, large field-of-view label-free imaging via aberration-corrected, closed-form complex field reconstruction

Computational imaging methods empower modern microscopes to produce high-resolution, large field-of-view, aberration-free images. Fourier ptychographic microscopy can increase the space-bandwidth product of conventional microscopy, but its iterative reconstruction methods are prone to parameter selection and tend to fail under excessive aberrations. Spatial Kramers–Kronig methods can analytically reconstruct complex fields, but is limited by aberration or providing extended resolution enhancement. Here, we present APIC, a closed-form method that weds the strengths of both methods while using only NA-matching and darkfield measurements. We establish an analytical phase retrieval framework which demonstrates the feasibility of analytically reconstructing the complex field associated with darkfield measurements. APIC can retrieve complex aberrations of an imaging system with no additional hardware and avoids iterative algorithms, requiring no human-designed convergence metrics while always obtaining a closed-form complex field solution. We experimentally demonstrate that APIC gives correct reconstruction results where Fourier ptychographic microscopy fails when constrained to the same number of measurements. APIC achieves 2.8 times faster computation using image tile size of 256 (length-wise), is robust against aberrations compared to Fourier ptychographic microscopy, and capable of addressing aberrations whose maximal phase difference exceeds 3.8π when using a NA 0.25 objective in experiment.

Effect of the span length for using finite to analogize infinite beam on viscoelastic foundation under a harmonic moving load

Railways with continuous rails are often modeled as infinite beams. In this paper, closed-form solution is firstly derived for the response of an infinite beam supported by viscoelastic foundation under a harmonic moving load by residue theorem method. The most suitable span length will be found for the analogic finite beam to best represent the deflection of the infinite beam for practical reasons. Starting from the equations of motion, closed-form solution is firstly derived for the infinite beam. Then, a FEM modeling procedure will be adopted to conform the validity of the solution derived analytically. The deflection curves of the infinite beam and analogic finite beam are compared for various parameters.

On the resonance frequency of the irregular shaped visco-acoustic Helmholtz cavity problem

In this paper we develop a closed-form mathematical solution for the resonance frequency of the axisymmetric irregular shaped visco-acoustic Helmholtz cavity problem. The solution is based on a conformal mapping that transforms the governing visco-acoustic equations of motion for the irregular cavity geometry into a new domain described by a set of orthogonal curvilinear coordinates. These orthogonal curvilinear coordinates have an important characteristic relative to the transformation of the physical boundary: the mapped domain is constructed such that the locus of all points for the length-wise profile of the physical boundary coincides with a constant value for a (quasi radial) coordinate in the new mapped domain. In these orthogonal curvilinear coordinates we obtain an analytical solution for the visco-acoustic equations of motion and exactly satisfy the no-slip fluid boundary conditions over the irregular shaped cavity walls by matching the coefficients of an orthogonal Legendre polynomial series representation for the non-uniform velocity on the cavity walls. The mathematical solution is an approximation to the extent that the scale factors chosen for the mapping consist of strictly the mutual-coupling terms of the true scale factors. Thus we analyze some results to understand the correlation of the analytical solution frequency response spectra with numerical finite element analyses of the idealized boundary conditions with various irregular shaped cavity geometries. The results for fundamental resonance frequency were found to exhibit good correlation with the results of finite element analyses over a significant range of fluid properties with different irregular cavity geometries. As the primary motivation for the derivation stems from the lack of a rapid yet accurate alternative to computationally intensive MultiPhysics FEA in the formulation of acoustic trendlines for sensor development, the solution’s acoustic impedance was applied as dispersive loading in a simple lumped parameter model of a fluid identification sensor for comparison. The resulting acoustic trendlines of the reduced (5 DOF) model were found to exhibit good correlation with the results of the MultiPhysics FEA (>105 DOF). Thus the solution was found to enable a closed-form analytical method for rapid construction of parametric studies in the practical design of fluid analysis sensors based on complex resonant cavity geometries.

Improved weighted nuclear norm with total variation for removing multiplicative noise

This paper introduces an improved weighted nuclear norm with a total variation model tailored for removing multiplicative noise. The model incorporates a weight matrix to regularize the residual matrix, effectively leveraging image redundancy to differentiate various statistical properties of the noise. Since there is no guarantee of a unique solution, the model is reformulated as a linear equality constraint problem and decomposed into two subproblems. These are addressed by using the alternating direction method of multipliers and the split Bregman method, respectively. In addition, each alternative update step has a closed-form and convergent solution. After obtaining the denoised image in the log-domain, the recovered image is given by using the exponential function and bias correction. Experimental evaluations demonstrate the efficacy of our algorithms in enhancing image restoration quality.

Analysis of incorporating Modified Weibull Model Fault Detection Rate Function into Software Reliability Modeling

When software systems are introduced, they are typically deployed in field environments similar to those used during development and testing. However, these systems may also be used in various other locations with different environmental conditions, making it challenging to improve software reliability. Factors such as the specific operating environment and the location of bugs in the code contribute to this difficulty. In this paper, we propose a new software reliability model that accounts for the uncertainty of operating environments. We present the explicit closed-form mean value function solution for the proposed model. The model's goodness of fit is demonstrated by comparing it to the nonhomogeneous Poisson process (NHPP) model based on Weibull model, using four sets of failure data sets from software applications. The proposed model performs well under various estimation techniques, making it a versatile tool for practitioners and researchers alike. The proposed model outperforms other existing NHPP Weibull based in terms of fitting accuracy under two different methods of estimation and provides a more detailed and precise evaluation of software reliability. Additionally, sensitivity analysis shows that the parameters of the suggested distribution significantly impact the mean value function.

Fourier Domain Robust Denoising Decomposition and Adaptive Patch MRI Reconstruction.

The sparsity of the Fourier transform domain has been applied to magnetic resonance imaging (MRI) reconstruction in k -space. Although unsupervised adaptive patch optimization methods have shown promise compared to data-driven-based supervised methods, the following challenges exist in MRI reconstruction: 1) in previous k -space MRI reconstruction tasks, MRI with noise interference in the acquisition process is rarely considered. 2) Differences in transform domains should be resolved to achieve the high-quality reconstruction of low undersampled MRI data. 3) Robust patch dictionary learning problems are usually nonconvex and NP-hard, and alternate minimization methods are often computationally expensive. In this article, we propose a method for Fourier domain robust denoising decomposition and adaptive patch MRI reconstruction (DDAPR). DDAPR is a two-step optimization method for MRI reconstruction in the presence of noise and low undersampled data. It includes the low-rank and sparse denoising reconstruction model (LSDRM) and the robust dictionary learning reconstruction model (RDLRM). In the first step, we propose LSDRM for different domains. For the optimization solution, the proximal gradient method is used to optimize LSDRM by singular value decomposition and soft threshold algorithms. In the second step, we propose RDLRM, which is an effective adaptive patch method by introducing a low-rank and sparse penalty adaptive patch dictionary and using a sparse rank-one matrix to approximate the undersampled data. Then, the block coordinate descent (BCD) method is used to optimize the variables. The BCD optimization process involves valid closed-form solutions. Extensive numerical experiments show that the proposed method has a better performance than previous methods in image reconstruction based on compressed sensing or deep learning.

Joint Anchor Graph Embedding and Discrete Feature Scoring for Unsupervised Feature Selection.

The success of existing unsupervised feature selection (UFS) methods heavily relies on the assumption that the intrinsic relationships among original high-dimensional (HD) data samples exist in the discriminative low-dimension (LD) subspace. However, previous UFS methods commonly construct pairwise graphs and employ l2,1 -norm regularization to severally preserve the local structure and calculate the score of features, which is computationally complex and easy to get stuck into local optimum, so that those approaches cannot be applied in dealing with large-scale datasets in practice. To overcome this challenge, we propose a novel UFS method, in which a novel anchor graph embedding paradigm is designed to extract the local neighborhood relationships among data samples by reducing the computational complexity of graph construction to be linear in the number of data. Moreover, to improve the optimality of selected features as well as the performance of downstream tasks, we propose a discrete feature scoring mechanism, which imposes orthogonal l2,0 -norm constraints on learned projections, in order to enhance the distinction of feature scores as well as reduce the probability of falling into local optimum. In addition, solving the proposed nonconvex and nonsmooth NP-hard problem is challenging, and we present an efficient optimization algorithm to address it and acquire a closed-form solution of the transformation matrix. Extensive experiments demonstrate the effectiveness and efficiency of the proposed UFS by comparison with several state-of-the-art approaches to clustering and image segmentation tasks.