Mathematical Formulation Research Articles

Mixed-integer linear programming for computing optimal experimental designs

The problem of computing an exact experimental design that is optimal for the least-squares estimation of the parameters of a regression model is considered. We show that this problem can be solved via mixed-integer linear programming (MILP) for a wide class of optimality criteria, including the criteria of A-, I-, G- and MV-optimality. This approach improves upon the current state-of-the-art mathematical programming formulation, which uses mixed-integer second-order cone programming. The key idea underlying the MILP formulation is McCormick relaxation, which critically depends on finite interval bounds for the elements of the covariance matrix of the least-squares estimator corresponding to an optimal exact design. We provide both analytic and algorithmic methods for constructing these bounds. We also demonstrate the unique advantages of the MILP approach, such as the possibility of incorporating multiple design constraints into the optimization problem, including constraints on the variances and covariances of the least-squares estimator.

Biomagnetic flow with magnetic particles over a continuously moving sheet affected by a magnetic dipole

This research concentrates on the 2-D, steady, laminar, viscous, incompressible boundary layer flow of a biomagnetic fluid containing two different magnetic particles (CoFe2O4andFe3O4) over a continuously moving horizontal plate in the presence of a magnetic field generated by a magnetic dipole. For the mathematical formulation the comprehensive concept of Biomagnetic Fluid Dynamics (BFD) is adopted incorporating the principles of FerroHydroDynamics (FHD) and MagnetoHydroDynamics (MHD). The physical problem which is constituted by a coupled system of Partial Differential Equations (PDEs) along with corresponding boundary conditions, is transformed into a coupled system of nonlinear Ordinary Differential Equations (ODEs) subject to analogous boundary conditions by establishing newly simplified similarity transformations. The transformed ODEs along with the boundary conditions are then solved numerically by introducing an efficient numerical technique based on a finite difference algorithm. Verification of this work has been also done by comparing the obtained results with previously published results and found in quite good agreement. The significant effects caused by the variation of the governing parameters such as the skin friction, heat transfer rate and wall pressure are presented more intricately. It has been contemplated that including magnetic particles with pure blood enhances the impact of the magnetic field on the flow, temperature and pressure profiles which could be of interest engineering implementations, like, magnetic drug delivering in blood cells, separating RBCs (Red Blood Cells), controlling the flow of blood during surgeries, treating cancer by producing magnetic hyperthermia etc.

Multivariate formulation to predict the frictional strength of fiber reinforced soils using gene expression programming

Modeling and prediction approaches have attracted much interest in recent years to evaluate the efficiency of fiber reinforcement in improving the strength characteristics of loose soils. Several analytical and artificial intelligence (AI) based models have been presented to estimate friction angle of fiber reinforced soils (FRS). However, due to limited input characteristics, these analytical models lack accuracy, whereas AI-based models are developed on advanced programming methods and need substantial programming expertise, resulting in a lack of agreement in mechanism and application. Herein, we provided a multivariate formulation that uses gene expression programming (GEP) to predict the frictional strength of FRS using fifteen influencing factors, which, to the best of our knowledge, has never been reported before. In this regard, two mathematical models are presented, one of which incorporates both qualitative and quantitative input factors, while the other just considers quantitative variables. Laboratory experimental data on FRS from various sources was assessed and properly analysed within the proposed framework while taking into account various affecting factors. The results illustrate the efficiency of GEP in estimating FRS frictional strength and emphasis the need of considering several parameters for prediction. Sensitivity and parametric studies were also performed to identify the relevance of each influencing factor and to determine whether the prediction models meet physical processes or are purely empirical relations. Finally, a user-friendly computer program was developed based on the GEP derived mathematical formulations to anticipate the FRS friction angle, which offers practitioners great field applicability.

Revisiting the Becker-Kistiakowsky-Wilson equation of state

The Jones-Wilkins-Lee (JWL) equation of state (EOS) is the most popular equation of state used in hydrocodes to model detonation products thermodynamics resulting from high explosives. However, the JWL EOS presents several major difficulties. Namely, its range of validity and convexity is limited as its parameters are adjusted on a given reference curve. When used in conditions where the thermodynamic state varies significantly, computational failure may happen. It occurs frequently in multiphase flow computations when heat and mass transfers with other phases are present, as the resulting thermodynamic state is far from the reference curve used for adjustment. Moreover, the detonation products composition is absent in the JWL formulation. This is another important limitation when additional physics are involved, such as post-combustion, phase transition, or when dealing with non-ideal explosives. In this work, the Becker-Kistiakowsky-Wilson (BKW) EOS is considered instead. The BKW EOS is widely used in thermochemical codes, as parent EOS to provide the reference data used to fit JWL EOS parameters. The BKW EOS present several advantages. Its formulation considers variable gas species in the detonation products. No specific adjustment is needed for a given explosive, and only the detonation products composition requires computational efforts. Its range of validity is large, as the various parameters have been adjusted for each gas species separately. However, the BKW EOS also present difficulties which limit its application in hydrocodes. These difficulties are the aim of the present work. The BKW EOS is currently restricted to thermochemical codes as its mathematical formulation is computationally expensive. It also requires the gas chemical composition to be computed via an appropriate chemical equilibrium solver. In the present work, the computational cost of the BKW EOS is reduced thanks to the thermodynamic relaxation method of Neron et al. (2023). The variable composition of the detonation products is accounted for through specific relaxation terms computed once, with the help of conventional thermochemical codes. The condensed species, such as solid carbon, are modelled with the Cochran-Chan (CC) EOS, having a simpler thermodynamic formulation and being thermodynamically consistent compared to usual EOSs used for condensed species. This multiphase formulation made of BKW gas mixture with variable composition and condensed phases is considered in temperature and pressure equilibrium. It is embedded in a multiphase flow formulation, with the help of the thermodynamic relaxation method mentioned above. It shows comparable computational cost as the JWL formulation, with significantly enhanced physics capabilities and improved robustness.

Review Paper on Machine Learning Algorithms

This paper comprehensively reviews widely used machine learning algorithms across supervised, unsupervised, and reinforcement learning paradigms. It covers linear models, decision trees, support vector machines, neural networks, clustering techniques, dimensionality reduction methods, and ensemble approaches. For each algorithm, theoretical foundations, mathematical formulations, practical considerations like parameter tuning and computational complexity, and real-world applications across domains like computer vision and finance are discussed. Challenges and limitations such as overfitting and scalability are explored. Recent advancements like deep learning and transfer learning are highlighted. Finally, a comparative analysis evaluating strengths, weaknesses, and suitable problem domains for the algorithms is provided, serving as a guide for effective utilization of machine learning techniques. Keywords:- Machine learning · Deep learning, Gradient Descent, Logistic Regression, Support Vector Machine, K Nearest Neighbor, Predictive analytics,

Improved fuzzy multi-objective transportation problem with Triangular fuzzy numbers

The present study investigates a Multi-Objective Transportation Problem within a fuzzy environment. The cost of transportation, supply, and demand data are assumed to be inaccurate due to market variations. As a result, the parameters are imprecise or fuzzy data. We offer a multi-objective, balanced transportation problem during this work, where all the parameters are fuzzy numbers. Following a mathematical formulation, fuzzy arithmetic will be used to divide the Fuzzy MOTP into three levels MOTP (lower, medium, upper). After reducing the problem to a crisp MOTP and applying a harmonic mean to each objective function, a suggested solution procedure is presented. Determining the optimal solutions for the FMOTP under unknown situations is, thus, the most important objective of this research.

Physics-informed neural networks for fully non-linear free surface wave propagation

This study proposed fully nonlinear free surface physics-informed neural networks (FNFS-PINNs), an advancement within the framework of PINNs, to tackle wave propagation in fully nonlinear potential flows with the free surface. Utilizing the nonlinear fitting capabilities of neural networks, FNFS-PINNs offer an approach to addressing the complexities of modeling nonlinear free surface flows, broadening the scope for applying PINNs to various wave propagation scenarios. The improved quasi-σ coordinate transformation and dimensionless formulation of the basic equations are adopted to transform the time-dependent computational domain into the stationary one and align variable scale changes across different dimensions, respectively. These innovations, alongside a specialized network structure and a two-stage optimization process, enhance the mathematical formulation of nonlinear water waves and solvability of the model. FNFS-PINNs are evaluated through three scenarios: solitary wave propagation featuring nonlinearity, regular wave propagation under high dispersion, and an inverse problem of nonlinear free surface flow focusing on the back-calculation of an initial state from its later state. These tests demonstrate the capability of FNFS-PINNs to compute the propagation of solitary and regular waves in the vertical two-dimensional scenarios. While focusing on two-dimensional wave propagation, this study lays the groundwork for extending FNFS-PINNs to other free surface flows and highlights their potential in solving inverse problems.

Inverse-Free DZNN Models for Solving Time-Dependent Linear System via High-Precision Linear Six-Step Method.

Time-dependent linear system (TDLS) is usually encountered in scientific research, which is the mathematical formulation of many practical applications. Different from conventional inverse-need models, by utilizing zeroing neural network (ZNN) method twice, an inverse-free continuous ZNN (CZNN) model is developed for solving TDLS. For conveniently practical use, a discrete model is naturally desired. Superior to conventional discretization methods, a general linear six-step (LSS) method with the seventh-order precision and five variable parameters is proposed for the first time. Constraints about five variable parameters are theoretically analyzed to guarantee the efficacy of the general LSS method. Within constraints, 12 specific LSS methods are further developed. Aided with the general LSS method, an inverse-free discrete ZNN (DZNN) is proposed and termed DZNN-LSS model, and its precision is greatly improved compared with conventional discrete models. For comparison, three conventional discretization methods are also utilized to generate DZNN models. Detailed theoretical analyses are provided to prove the efficacy of relevant models. In addition, a specific TDLS example is considered to show the effectiveness and superiority of the DZNN-LSS model. More than that, applications to manipulator control and sound source localization are conducted to illustrate the applicability of the DZNN-LSS model.

Comparison of steady-state analytical wake models implemented in wind farm analysis software

A common set of mathematical wind turbine wake models are implemented in a few, well-adopted computational tools for wind farm wake modelling. Although the referenced mathematical formulations are common, implementation details may lead to differences in results. This study presents a systematic comparison of the implementation of mathematical wake models in open source, Python-based wind turbine wake modelling software, and a set of the models are directly compared. Despite aligning only the mathematical model parameters and retaining the default computational model parameters, good agreement is found across most of the model implementations, and additional agreement is expected upon further parameters alignment.

Supplier selection in green supply chain management using correlation-based TOPSIS in a q-rung orthopair fuzzy soft environment

Fuzzy hybrid models are efficient mathematical tools for managing unclear and vague data in real-world scenarios. This research explores the q-rung orthopair fuzzy soft set (q-ROFSS), which presents incomplete and ambiguous details in decision-making problems. The main intention of this study is to describe and evaluate the characteristics of the correlation coefficient (CC) and weighted correlation coefficient (WCC) for q-ROFSS. Also, the technique for order preference should be enhanced by similarity to the ideal solution (TOPSIS) with extended measures in q-ROFSS settings. Furthermore, we integrated mathematical formulations of correlation obstructions to confirm the consistency of the planned technique. It helps handle difficulties involving multi-attribute group decision-making (MAGDM). Moreover, a numerical illustration is presented to clarify how the advocated decision-making methodology can be implemented in evaluating suppliers in green supply chain management (GSCM). As a result, each alternative is assessed using multiple criteria, such as quality and reliability, capacity and scalability, compliance and certifications, and sustainability practices. The technique proposed in this study retains the selected research's specific structure more effectively than current techniques. A comparative analysis further substantiates the feasibility and effectiveness of the proposed approach over other decision-making techniques.

Two moments preserving sectional approach for an enzymatic coagulation equation

The coagulation process has found extensive applications in monitoring the evolution of aerosol and granule preparation in pharmaceutical sciences, blood clotting in biology, and cheese manufacturing due to the enzymes in the dairy industry. Among these, modeling the cheese manufacturing process is more challenging due to three indistinguishable sub-mechanisms: (a) enzymatic proteolysis, (b) coagulation, and (c) gelation, which occurs during the enzymatic coagulation of milk. The current study focuses on developing a sectional approach based on the cell average technique for monitoring the evolution of enzyme-induced coagulation of paracasein micelles over time. The proposed technique preserves two integral properties, such as total number and total volume in the system. The mathematical formulation of the proposed technique is very simple, easy to code, and has a robust implementation on any uniform and non-uniform grids. Due to the unavailability of the analytical solutions of the number density functions, the validation of the new proposed approach is done by extracting the new series solutions through the modification of the Homotopy perturbation method [Kaur et al., J. Phys. A 52(38), 385201 (2019)] and exact integral moments for several kernels. It has been shown that the new approach not only estimates the first two integral moments accurately but also computes the second-order moment with high precision without any specific measures. Moreover, domains of varying size grids are taken into account to analyze the convergence behavior of the average-size paracasein micelles formed in the system based on the zeroth and first moments.

The impact of circular economy strategies on municipal waste management: A system dynamics approach

The low levels of reuse and recycling of solid waste in cities in developing countries can be attributed to decision-making processes with sub investment that do not consider waste streams, recovery capacities, and potential for recycling. This is due to a lack of understanding of the systemic structure of urban waste management.This article presents the development of a model of an urban waste management system that integrates decisions based on strategies that close cycles, belonging to the concept of a circular economy. These strategies aim to increase the potential use of organic waste, plastics, and glass.The model was built using the Systems Dynamics methodology, which allows obtaining mathematical models for the study and observation of trends and future scenarios of complex systems. The model is made up of three modules: health, financial, and circular economy. The strategies evaluated and compared in the simulation environment were: Economic incentives at the price of the material used; Increase in waste sorting capacity; Increase in the capacity to use waste.Methodologically, novel mathematical representations were achieved for the estimation of the useful life of sanitary landfills. Likewise, a mathematical formulation was found for the automatic evaluation of the efficiency in the capacity to use different wastes, which allows to know the need for investment or increase of capacity to use the type of waste evaluated.Using novel mathematical representations, this study achieved a methodological breakthrough in estimating the useful life of sanitary landfills and waste transformation efficiency. These representations also enable the automatic evaluation of how effectively different types of waste can be utilized. This information can then be used to determine the need for investment in, or expansion of, the capacity to handle specific waste types or to increase the overall waste collection capacity of the city.

Optimisation of process plan generation for reconfigurable manufacturing systems: efficient heuristics and lower bounds

ABSTRACT We address the process plan generation problem in a reconfigurable manufacturing environment. The objective is to minimise the total production time of a given part which includes the processing times of operations, as well as the changeover times for machines, configurations, and tools. A new efficient 0–1 mathematical programming formulation is proposed, together with a heuristic and two lower bounds. The formulation is based on one main decision variable (1MDV). It outperforms a well known two-main decision variable formulation (2MDV) from the literature. Using a commercial solver, the 1MDV formulation is on average almost five times faster than the 2MDV formulation across all instances where optimality was achieved. The proposed heuristic is a decomposition mathematical programming-based heuristic. For small benchmark instances solved to optimality using 1MDV formulation, the heuristic obtains solutions at less than 1% distance on average from optimum in few seconds. The best of the two proposed lower bounds, LB2, is obtained in less than 0.56 seconds, across all tested instances, while the 1MDV model needs about 30 seconds on average to reach the same result. These lower bounds are used to measure the performance of the heuristics on large size instances

Resolution of the Fundamental Equation of the Mathematical Formulation of Nuclear Collision Mechanisms Model

The fundamental equation of the model titled “mathematical formulation of nuclear collision mechanisms” [1] was solved. The solution is compounded of two steps, the first step consists in calculating the values of the parameters of the mechanism-operator underlying a given nuclear collision and the second step, consists in deducing the open channels for this collision. Knowing that the equation is not classic, its resolution has required the development of an innovative mathematical method that sets on the use of physical constraints of the problem.

PROJECT PROPOSAL: REJUVENATION OF BRAIN NEURONS USING QUANTUM ENTANGLEMENT IN NUCLEOTIDES AND CRYPTOCHROMES

In this paper, we propose a novel and futuristic theoretical framework aimed at rejuvenating and repairing human brain neurons through the application of quantum entanglement. We hypothesize that by inducing atomic entanglement in nucleotides and photonic entanglement in cryptochromes, and subsequently reinjecting these entangled particles into brain neurons, we can exploit the unique properties of quantum entanglement to enhance neuronal repair and regeneration. Our study delves into the underlying quantum mechanisms, presents detailed mathematical formulations, and outlines comprehensive experimental protocols for achieving neuronal rejuvenation. Additionally, we explore the potential amplification of these effects by exposing entangled particles to the space environment, leveraging factors such as microgravity, cosmic radiation, vacuum conditions, temperature extremes, and electromagnetic isolation. This approach is specifically tailored for brain neurons, as other cells in the body do not process and transmit information in a manner that would benefit from quantum entanglement. Furthermore, we discuss the potential application of this method in treating blindness and neurodegenerative diseases such as Alzheimers disease.

A Numerical Model to Predict the Relaxation Phenomena in Thermoset Polymers and Their Effects on Residual Stress during Curing, Part II: Numerical Evaluation of Residual Stress.

This article proposes a numerical routine to predict the residual stresses developing in an epoxy component during its curing. The scaling of viscoelastic properties with the temperature and the degree of conversion is modeled, adopting a mathematical formulation that considers the concurrent effects of curing and structural relaxation on the epoxy's viscoelastic relaxation time. The procedure comprises two moduli: at first, the thermal-kinetical problem is solved using the thermal module of Ansys and a homemade routine written in APDL, then the results in terms of temperature and the degree of conversion profiles are used to evaluate the viscoelastic functions, and the structural problem is solved in the mechanical module of Ansys, allowing the residual stresses calculation. The results show that the residual stresses mainly arise during cooling and scale with the logarithm of the Biot number.

DRD-GAN: A Novel Distributed Conditional Wasserstein Deep Convolutional Relativistic Discriminator GAN with Improved Convergence

Generative Adversarial Network (GAN) exhibited significant capabilities in many applications including image enhancement and manipulation, language translation, generating images/videos from text, creating art and music, and so on. However, train ing GANs using large datasets remains highly computationally intensive for most of the standalone systems. Additionally, standalone GANs often exhibit poor synchronization between their generator and discriminator with unstable training , poor convergence along with a large number of mode collapse s and vanishing / exploding gradients. Standalone GANs also failed to learn in a decentralized environment, where the data is distributed among several client machines. Some researchers have lately used the most prevalent decentralized setting available today, called Federated Learning (FL) to develop distributed -GAN strategies as the possible solutions, although their implementations mostly failed to address the above issues mainly because of: the training instability within the distributed component s, which eventually leads to the poor synchronization among the generator s and discriminator s scattered over several machines. In this work, we developed a computationally inexpensive Wasserstein conditional Distributed Relativistic Discriminator-GAN or DRD-GAN to alleviate the above issues. DRD-GAN stabilizes its train ing (with non-convex losses) by keeping a global generator in the central server and relativistic discriminator s in the local client s (one discriminator per client ), and uses Wasserstein-1 for computing local and global losses. It eventually avoids mode collapse s, vanishing/exploding gradient s (both in the presence of iid and non-iid samples) and helps DRD-GAN to produce high-quality fake images. Apart from that, the sheer unavailability of a capable conditional distributed -GAN model has become another motivation behind the current work. Essentially, we revisited the FL paradigms, locating one discriminator per client , and a generator in the central server that aggregates the updates coming from multiple discriminator s. Relativistic discriminator s in the client s are train ed on both iid and non-iid private data. We presented a detailed mathematical formulation of DRD-GAN and empirically evaluated our claims using CIFAR-10, MNIST, EuroSAT, and CelebA datasets.

Cardiac length-dependent activation driven by force-dependent thick-filament dynamics

The length-dependent activation (LDA) of maximum force and calcium sensitivity are established features of cardiac muscle contraction but the dominant underlying mechanisms remain to be fully clarified. Alongside the well-documented regulation of contraction via the thin filaments, experiments have identified an additional force-dependent thick-filament activation, whereby myosin heads parked in a so-called off state become available to generate force. This process produces a feedback effect that may potentially drive LDA. Using biomechanical modeling of a human left-ventricular myocyte, this study investigates the extent to which the off-state dynamics could, by itself, plausibly account for LDA, depending on the specific mathematical formulation of the feedback. We hypothesized four different models of the off-state regulatory feedback based on (A) total force, (B) active force, (C) sarcomere strain, and (D) passive force. We tested if these models could reproduce the isometric steady-state and dynamic LDA features predicted by an earlier published model of a human left-ventricle myocyte featuring purely phenomenological length dependences. The results suggest that only total-force feedback (A) is capable of reproducing the expected behaviors, but that passive tension could provide a length-dependent signal on which to initiate the feedback. Furthermore, by attributing LDA to off-state dynamics, our proposed model also qualitatively reproduces experimentally observed effects of the off-state-stabilizing drug mavacamten. Taken together, these results support off-state dynamics as a plausible primary mechanism underlying LDA.

Higher-order phase-space moments for off-diagonal rotating Morse oscillators

The present paper proposes an advancement beyond the recent mathematical formulation [Cherroud O and Yahiaoui S-A 2023 Eur. Phys. J. Plus. 138 534], by extending their approach to the evaluation of off-diagonal matrix elements 〈xˆmpˆl〉n,n′J,J′ for the rotation-vibration Morse oscillators. Calculations are conducted within the framework of phase-space quantum mechanics, considering (r → r eq). This methodology facilitates the derivation of analytical expressions for 〈xˆs〉n,n′J,J′ and 〈pˆs〉n,n′J,J′ matrix elements, where s = 1, 2, …. This approach enables the evaluation of both diagonal ( n=n′,J=J′≠0 ) and off-diagonal ( n≠n′,J=J′≠0 ) matrix elements through a more explicit and compact analytic formula, applicable for cases where J≠J′ as well as for scenarios involving non-rotating effects J=J′=0 .

PyHySCO: GPU-enabled susceptibility artifact distortion correction in seconds.

Over the past decade, reversed gradient polarity (RGP) methods have become a popular approach for correcting susceptibility artifacts in echo-planar imaging (EPI). Although several post-processing tools for RGP are available, their implementations do not fully leverage recent hardware, algorithmic, and computational advances, leading to correction times of several minutes per image volume. To enable 3D RGP correction in seconds, we introduce PyTorch Hyperelastic Susceptibility Correction (PyHySCO), a user-friendly EPI distortion correction tool implemented in PyTorch that enables multi-threading and efficient use of graphics processing units (GPUs). PyHySCO uses a time-tested physical distortion model and mathematical formulation and is, therefore, reliable without training. An algorithmic improvement in PyHySCO is its use of the one-dimensional distortion correction method by Chang and Fitzpatrick to initialize the non-linear optimization. PyHySCO is published under the GNU public license and can be used from the command line or its Python interface. Our extensive numerical validation using 3T and 7T data from the Human Connectome Project suggests that PyHySCO can achieve accuracy comparable to that of leading RGP tools at a fraction of the cost. We also validate the new initialization scheme, compare different optimization algorithms, and test the algorithm on different hardware and arithmetic precisions.