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15 pages, 2005 KiB

Open AccessArticle

An Innovative Mathematical Model of the Spine: Predicting Cobb and Intervertebral Angles Using the 3D Position of the Spinous Processes Measured by Vertebral Metrics

by Ana Teresa Gabriel, Cláudia Quaresma and Pedro Vieira

Algorithms 2024, 17(4), 134; https://doi.org/10.3390/a17040134 - 25 Mar 2024

Viewed by 1140

Abstract

Back pain is regularly associated with biomechanical changes in the spine. The traditional methods to assess spine biomechanics use ionising radiation. Vertebral Metrics (VM) is a non-invasive instrument developed by the authors in previous research that assesses the spinous processes’ position. However, the [...] Read more.

Back pain is regularly associated with biomechanical changes in the spine. The traditional methods to assess spine biomechanics use ionising radiation. Vertebral Metrics (VM) is a non-invasive instrument developed by the authors in previous research that assesses the spinous processes’ position. However, the spine model used by VM is not accurate. To overcome it, the present paper proposes a pioneering and simple articulated model of the spine built through the data collected by VM. The model is based on the spring–mass system and uses the Levenberg–Marquardt algorithm to find the arrangement of vertebral bodies. It represents the spine as rigid geometric transformations from one vertebra to the other when the extremity vertebrae are stationary. The validation process used the Bland–Altman method to compare the Cobb and the intervertebral angles computed by the model with the radiographic exams of eight patients diagnosed with Ankylosing Spondylitis. The results suggest that the model is valid; however, previous clinical information would improve outcomes by customising the lower and upper vertebrae positions, since the study revealed that the C6 rotation slightly influences the computed angles. Applying VM with the new model could make a difference in preventing, monitoring, and early diagnosing spinal disorders. Full article

(This article belongs to the Special Issue Mathematical Modelling in Engineering and Human Behaviour)

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12 pages, 1526 KiB

Open AccessArticle

An Efficient Third-Order Scheme Based on Runge–Kutta and Taylor Series Expansion for Solving Initial Value Problems

by Noori Y. Abdul-Hassan, Zainab J. Kadum and Ali Hasan Ali

Algorithms 2024, 17(3), 123; https://doi.org/10.3390/a17030123 - 16 Mar 2024

Viewed by 1346

Abstract

In this paper, we propose a new numerical scheme based on a variation of the standard formulation of the Runge–Kutta method using Taylor series expansion for solving initial value problems (IVPs) in ordinary differential equations. Analytically, the accuracy, consistency, and absolute stability of [...] Read more.

In this paper, we propose a new numerical scheme based on a variation of the standard formulation of the Runge–Kutta method using Taylor series expansion for solving initial value problems (IVPs) in ordinary differential equations. Analytically, the accuracy, consistency, and absolute stability of the new method are discussed. It is established that the new method is consistent and stable and has third-order convergence. Numerically, we present two models involving applications from physics and engineering to illustrate the efficiency and accuracy of our new method and compare it with further pertinent techniques carried out in the same order. Full article

(This article belongs to the Special Issue Mathematical Modelling in Engineering and Human Behaviour)

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13 pages, 428 KiB

Open AccessArticle

A Geometrical Study about the Biparametric Family of Anomalies in the Elliptic Two-Body Problem with Extensions to Other Families

by José Antonio López Ortí, Francisco José Marco Castillo and María José Martínez Usó

Algorithms 2024, 17(2), 66; https://doi.org/10.3390/a17020066 - 4 Feb 2024

Viewed by 1608

Abstract

In the present paper, we efficiently solve the two-body problem for extreme cases such as those with high eccentricities. The use of numerical methods, with the usual variables, cannot maintain the perihelion passage accurately. In previous articles, we have verified that this problem [...] Read more.

In the present paper, we efficiently solve the two-body problem for extreme cases such as those with high eccentricities. The use of numerical methods, with the usual variables, cannot maintain the perihelion passage accurately. In previous articles, we have verified that this problem is treated more adequately through temporal reparametrizations related to the mean anomaly through the partition function. The biparametric family of anomalies, with an appropriate partition function, allows a systematic study of these transformations. In the present work, we consider the elliptical orbit as a meridian section of the ellipsoid of revolution, and the partition function depends on two variables raised to specific parameters. One of the variables is the mean radius of the ellipsoid at the secondary, and the other is the distance to the primary. One parameter regulates the concentration of points in the apoapsis region, and the other produces a symmetrical displacement between the polar and equatorial regions. The three most used geodesy latitude variables are also studied, resulting in one not belonging to the biparametric family. However, it is in the one introduced now, which implies an extension of the biparametric method. The results obtained using the method presented here now allow a causal interpretation of the operation of numerous reparametrizations used in the study of orbital motion. Full article

(This article belongs to the Special Issue Mathematical Modelling in Engineering and Human Behaviour)

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22 pages, 3538 KiB

Open AccessArticle

Analysis of the Effectiveness of a Freight Transport Vehicle at High Speed in a Vacuum Tube (Hyperloop Transport System)

by David S. Pellicer and Emilio Larrodé

Algorithms 2024, 17(1), 17; https://doi.org/10.3390/a17010017 - 29 Dec 2023

Viewed by 1989

Abstract

This paper shows the development of a numerical analysis model, which enables the calculation of the cargo transport capacity of a vehicle that circulates through a vacuum tube at high speed, whose effectiveness in transport is analyzed. The simulated transportation system is based [...] Read more.

This paper shows the development of a numerical analysis model, which enables the calculation of the cargo transport capacity of a vehicle that circulates through a vacuum tube at high speed, whose effectiveness in transport is analyzed. The simulated transportation system is based on vehicles moving in vacuum tubes at high speed, a concept commonly known as Hyperloop, but assuming the vehicles for cargo containers. For the specific vehicle proposed, which does not include a compressor and levitates on magnets, the system formed by the vehicle and the vacuum tube has been conceptually developed, establishing the corresponding mathematical relationships that define its behavior. To properly model the performance of this transport system, it has been necessary to establish the relationships between the design variables and the associated constraints, such as the Kantrowitz limit, aerodynamics, transport, energy consumption, etc. Once the model was built and validated, it was used to analyze the effects of the variation of the number of containers, the operating speed and the tube length, considering the total and specific consumption of energy. After finding the most efficient configuration regarding energy consumption and transport effectiveness, the complete system was calculated. The results obtained constitute a first approximation for the predesign of this transport system and the built model allows different alternatives to be compared according to the design variables. Full article

(This article belongs to the Special Issue Mathematical Modelling in Engineering and Human Behaviour)

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16 pages, 4407 KiB

Open AccessArticle

Predicting Pedestrian Trajectories with Deep Adversarial Networks Considering Motion and Spatial Information

by Liming Lao, Dangkui Du and Pengzhan Chen

Algorithms 2023, 16(12), 566; https://doi.org/10.3390/a16120566 - 12 Dec 2023

Cited by 2 | Viewed by 1828

Abstract

This paper proposes a novel prediction model termed the social and spatial attentive generative adversarial network (SSA-GAN). The SSA-GAN framework utilizes a generative approach, where the generator employs social attention mechanisms to accurately model social interactions among pedestrians. Unlike previous methodologies, our model [...] Read more.

This paper proposes a novel prediction model termed the social and spatial attentive generative adversarial network (SSA-GAN). The SSA-GAN framework utilizes a generative approach, where the generator employs social attention mechanisms to accurately model social interactions among pedestrians. Unlike previous methodologies, our model utilizes comprehensive motion features as query vectors, significantly enhancing predictive performance. Additionally, spatial attention is integrated to encapsulate the interactions between pedestrians and their spatial context through semantic spatial features. Moreover, we present a novel approach for generating simulated multi-trajectory datasets using the CARLA simulator. This method circumvents the limitations inherent in existing public datasets such as UCY and ETH, particularly when evaluating multi-trajectory metrics. Our experimental findings substantiate the efficacy of the proposed SSA-GAN model in capturing the nuances of pedestrian interactions and providing accurate multimodal trajectory predictions. Full article

(This article belongs to the Special Issue Mathematical Modelling in Engineering and Human Behaviour)

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19 pages, 8484 KiB

Open AccessArticle

A Recommendation System Supporting the Implementation of Sustainable Risk Management Measures in Airport Operations

by Silvia Carpitella, Bruno Brentan, Antonella Certa and Joaquín Izquierdo

Algorithms 2023, 16(11), 511; https://doi.org/10.3390/a16110511 - 7 Nov 2023

Cited by 2 | Viewed by 2362

Abstract

This paper introduces a recommendation system aimed at enhancing the sustainable process of risk management within airport operations, with a special focus on Occupational Stress Risks (OSRs). The recommendation system is implemented via a flexible Python code that offers seamless integration into various [...] Read more.

This paper introduces a recommendation system aimed at enhancing the sustainable process of risk management within airport operations, with a special focus on Occupational Stress Risks (OSRs). The recommendation system is implemented via a flexible Python code that offers seamless integration into various operational contexts. It leverages Fuzzy Cognitive Maps (FCMs) to conduct comprehensive risk assessments, subsequently generating prioritized recommendations for predefined risk management measures aimed at preventing and/or reducing the most critical OSRs. The system’s reliability has been validated by iterating the procedure with diverse input data (i.e., matrices of varying sizes) and measures. This confirms the system’s effectiveness across a broad spectrum of engineering scenarios. Full article

(This article belongs to the Special Issue Mathematical Modelling in Engineering and Human Behaviour)

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14 pages, 1095 KiB

Open AccessArticle

A New Third-Order Family of Multiple Root-Findings Based on Exponential Fitted Curve

by Vinay Kanwar, Alicia Cordero, Juan R. Torregrosa, Mithil Rajput and Ramandeep Behl

Algorithms 2023, 16(3), 156; https://doi.org/10.3390/a16030156 - 12 Mar 2023

Cited by 2 | Viewed by 1915

Abstract

In this paper, we present a new third-order family of iterative methods in order to compute the multiple roots of nonlinear equations when the multiplicity

(m \geq 1)

is known in advance. There is a plethora of third-order point-to-point methods, available [...] Read more.

In this paper, we present a new third-order family of iterative methods in order to compute the multiple roots of nonlinear equations when the multiplicity

(m \geq 1)

is known in advance. There is a plethora of third-order point-to-point methods, available in the literature; but our methods are based on geometric derivation and converge to the required zero even though derivative becomes zero or close to zero in vicinity of the required zero. We use the exponential fitted curve and tangency conditions for the development of our schemes. Well-known Chebyshev, Halley, super-Halley and Chebyshev–Halley are the special members of our schemes for

m = 1

. Complex dynamics techniques allows us to see the relation between the element of the family of iterative schemes and the wideness of the basins of attraction of the simple and multiple roots, on quadratic polynomials. Several applied problems are considered in order to demonstrate the performance of our methods and for comparison with the existing ones. Based on the numerical outcomes, we deduce that our methods illustrate better performance over the earlier methods even though in the case of multiple roots of high multiplicity. Full article

(This article belongs to the Special Issue Mathematical Modelling in Engineering and Human Behaviour)

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Review

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34 pages, 581 KiB

Open AccessReview

Tensor-Based Approaches for Nonlinear and Multilinear Systems Modeling and Identification

by Gérard Favier and Alain Kibangou

Algorithms 2023, 16(9), 443; https://doi.org/10.3390/a16090443 - 14 Sep 2023

Cited by 1 | Viewed by 1848

Abstract

Nonlinear (NL) and multilinear (ML) systems play a fundamental role in engineering and science. Over the last two decades, active research has been carried out on exploiting the intrinsically multilinear structure of input–output signals and/or models in order to develop more efficient identification [...] Read more.

Nonlinear (NL) and multilinear (ML) systems play a fundamental role in engineering and science. Over the last two decades, active research has been carried out on exploiting the intrinsically multilinear structure of input–output signals and/or models in order to develop more efficient identification algorithms. This has been achieved using the notion of tensors, which are the central objects in multilinear algebra, giving rise to tensor-based approaches. The aim of this paper is to review such approaches for modeling and identifying NL and ML systems using input–output data, with a reminder of the tensor operations and decompositions needed to render the presentation as self-contained as possible. In the case of NL systems, two families of models are considered: the Volterra models and block-oriented ones. Volterra models, frequently used in numerous fields of application, have the drawback to be characterized by a huge number of coefficients contained in the so-called Volterra kernels, making their identification difficult. In order to reduce this parametric complexity, we show how Volterra systems can be represented by expanding high-order kernels using the parallel factor (PARAFAC) decomposition or generalized orthogonal basis (GOB) functions, which leads to the so-called Volterra–PARAFAC, and Volterra–GOB models, respectively. The extended Kalman filter (EKF) is presented to estimate the parameters of a Volterra–PARAFAC model. Another approach to reduce the parametric complexity consists in using block-oriented models such as those of Wiener, Hammerstein and Wiener–Hammerstein. With the purpose of estimating the parameters of such models, we show how the Volterra kernels associated with these models can be written under the form of structured tensor decompositions. In the last part of the paper, the notion of tensor systems is introduced using the Einstein product of tensors. Discrete-time memoryless tensor-input tensor-output (TITO) systems are defined by means of a relation between an Nth-order tensor of input signals and a Pth-order tensor of output signals via a

(P + N)

th-order transfer tensor. Such systems generalize the standard memoryless multi-input multi-output (MIMO) system to the case where input and output data define tensors of order higher than two. The case of a TISO system is then considered assuming the system transfer is a rank-one Nth-order tensor viewed as a global multilinear impulse response (IR) whose parameters are estimated using the weighted least-squares (WLS) method. A closed-form solution is proposed for estimating each individual IR associated with each mode-n subsystem. Full article

(This article belongs to the Special Issue Mathematical Modelling in Engineering and Human Behaviour)

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