Mathematics

14 pages, 1461 KiB

Open AccessArticle

Prompt Optimization in Large Language Models

by Antonio Sabbatella, Andrea Ponti, Ilaria Giordani, Antonio Candelieri and Francesco Archetti

Mathematics 2024, 12(6), 929; https://doi.org/10.3390/math12060929 - 21 Mar 2024

Viewed by 4957

Prompt optimization is a crucial task for improving the performance of large language models for downstream tasks. In this paper, a prompt is a sequence of n-grams selected from a vocabulary. Consequently, the aim is to select the optimal prompt concerning a certain [...] Read more.

Prompt optimization is a crucial task for improving the performance of large language models for downstream tasks. In this paper, a prompt is a sequence of n-grams selected from a vocabulary. Consequently, the aim is to select the optimal prompt concerning a certain performance metric. Prompt optimization can be considered as a combinatorial optimization problem, with the number of possible prompts (i.e., the combinatorial search space) given by the size of the vocabulary (i.e., all the possible n-grams) raised to the power of the length of the prompt. Exhaustive search is impractical; thus, an efficient search strategy is needed. We propose a Bayesian Optimization method performed over a continuous relaxation of the combinatorial search space. Bayesian Optimization is the dominant approach in black-box optimization for its sample efficiency, along with its modular structure and versatility. We use BoTorch, a library for Bayesian Optimization research built on top of PyTorch. Specifically, we focus on Hard Prompt Tuning, which directly searches for an optimal prompt to be added to the text input without requiring access to the Large Language Model, using it as a black-box (such as for GPT-4 which is available as a Model as a Service). Albeit preliminary and based on “vanilla” Bayesian Optimization algorithms, our experiments with RoBERTa as a large language model, on six benchmark datasets, show good performances when compared against other state-of-the-art black-box prompt optimization methods and enable an analysis of the trade-off between the size of the search space, accuracy, and wall-clock time. Full article

(This article belongs to the Special Issue Mathematical Methods in Machine Learning and Data Science)

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9 pages, 440 KiB

Open AccessArticle

Existence of Kink and Antikink Wave Solutions of Singularly Perturbed Modified Gardner Equation

by Weifang Yan, Linlin Wang and Min Zhang

Mathematics 2024, 12(6), 928; https://doi.org/10.3390/math12060928 - 21 Mar 2024

Viewed by 972

Abstract

In this paper, the singularly perturbed modified Gardner equation is considered. Firstly, for the unperturbed equation, under certain parameter conditions, we obtain the exact expressions of kink wave solution and antikink wave solution by using the bifurcation method of dynamical systems. Then, the [...] Read more.

In this paper, the singularly perturbed modified Gardner equation is considered. Firstly, for the unperturbed equation, under certain parameter conditions, we obtain the exact expressions of kink wave solution and antikink wave solution by using the bifurcation method of dynamical systems. Then, the persistence of the kink and antikink wave solutions of the perturbed modified Gardner equation is studied by exploiting the geometric singular perturbation theory and the Melnikov function method. When the perturbation parameter is sufficiently small, we obtain the sufficient conditions to guarantee the existence of kink and antikink wave solutions. Full article

(This article belongs to the Special Issue Numerical Methods for Partial Differential Equation)

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12 pages, 262 KiB

Open AccessEditor’s ChoiceArticle

A Generalized Hierarchy of Combined Integrable Bi-Hamiltonian Equations from a Specific Fourth-Order Matrix Spectral Problem

by Wen-Xiu Ma

Mathematics 2024, 12(6), 927; https://doi.org/10.3390/math12060927 - 21 Mar 2024

Cited by 7 | Viewed by 921

Abstract

The aim of this paper is to analyze a specific fourth-order matrix spectral problem involving four potentials and two free nonzero parameters and construct an associated integrable hierarchy of bi-Hamiltonian equations within the zero curvature formulation. A hereditary recursion operator is explicitly computed, [...] Read more.

The aim of this paper is to analyze a specific fourth-order matrix spectral problem involving four potentials and two free nonzero parameters and construct an associated integrable hierarchy of bi-Hamiltonian equations within the zero curvature formulation. A hereditary recursion operator is explicitly computed, and the corresponding bi-Hamiltonian formulation is established by the so-called trace identity, showing the Liouville integrability of the obtained hierarchy. Two illustrative examples are novel generalized combined nonlinear Schrödinger equations and modified Korteweg–de Vries equations with four components and two adjustable parameters. Full article

25 pages, 5565 KiB

Open AccessArticle

Predicting Fan Attendance at Mega Sports Events—A Machine Learning Approach: A Case Study of the FIFA World Cup Qatar 2022

by Ahmad Al-Buenain, Mohamed Haouari and Jithu Reji Jacob

Mathematics 2024, 12(6), 926; https://doi.org/10.3390/math12060926 - 21 Mar 2024

Cited by 1 | Viewed by 1974

Abstract

Mega sports events generate significant media coverage and have a considerable economic impact on the host cities. Organizing such events is a complex task that requires extensive planning. The success of these events hinges on the attendees’ satisfaction. Therefore, accurately predicting the number [...] Read more.

Mega sports events generate significant media coverage and have a considerable economic impact on the host cities. Organizing such events is a complex task that requires extensive planning. The success of these events hinges on the attendees’ satisfaction. Therefore, accurately predicting the number of fans from each country is essential for the organizers to optimize planning and ensure a positive experience. This study aims to introduce a new application for machine learning in order to accurately predict the number of attendees. The model is developed using attendance data from the FIFA World Cup (FWC) Russia 2018 to forecast the FWC Qatar 2022 attendance. Stochastic gradient descent (SGD) was found to be the top-performing algorithm, achieving an R² metric of 0.633 in an Auto-Sklearn experiment that considered a total of 2523 models. After a thorough analysis of the result, it was found that team qualification has the highest impact on attendance. Other factors such as distance, number of expatriates in the host country, and socio-geopolitical factors have a considerable influence on visitor counts. Although the model produces good results, with ML it is always recommended to have more data inputs. Therefore, using previous tournament data has the potential to increase the accuracy of the results. Full article

(This article belongs to the Special Issue Application of Machine Learning and Data Mining)

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14 pages, 3166 KiB

Open AccessArticle

Mathematical Modeling of Multi-Phenomena Anisotropic Systems: Ejection of Primary Aerosols during the Fast Pyrolysis of Biomass

by Mario A. Sánchez, Juan C. Maya, Farid Chejne, Brennan Pecha and Adriana M. Quinchía-Figueroa

Mathematics 2024, 12(6), 925; https://doi.org/10.3390/math12060925 - 21 Mar 2024

Viewed by 1182

Abstract

This study introduces a novel particle model for biomass fast pyrolysis, incorporating an anisotropic cylindrical particle to address mass and energy transport coupled with aerosol ejection, which previous models have overlooked. The main contribution lies in developing a model that considers aerosol generation [...] Read more.

This study introduces a novel particle model for biomass fast pyrolysis, incorporating an anisotropic cylindrical particle to address mass and energy transport coupled with aerosol ejection, which previous models have overlooked. The main contribution lies in developing a model that considers aerosol generation in anisotropic cylindrical particles for the first time, addressing bubbling dynamics and bursting within the liquid phase. The population balance equation describes bubble dynamics and aerosol formation, capturing phenomena like nucleation, growth, coalescence, and bursting. The model employs the method of moments with bubble volume as an internal variable, substantially reducing computational costs by eliminating dependence on this variable. Results highlight the significant impact of anisotropy and particle size on aerosol ejection: smaller, less elongated particles experience faster heating, quicker conversion, and the increased accumulation of the liquid intermediate phase. Specifically, 1 mm diameter particles yield higher concentrations of metaplast and bio-oil aerosols, exceeding 15%, compared to concentrations below 11% for 3 mm particles. This model provides insights into aerosol structure (volume, surface area), aiding in understanding aerosol reactivity at the reactor scale. Full article

(This article belongs to the Special Issue Computational Mathematics and Mathematical Modelling)

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14 pages, 284 KiB

Open AccessArticle

On Soft ω_δ-Open Sets and Some Decomposition Theorems

by Dina Abuzaid, Samer Al-Ghour and Monia Naghi

Mathematics 2024, 12(6), 924; https://doi.org/10.3390/math12060924 - 21 Mar 2024

Viewed by 883

Abstract

In this paper, we present a novel family of soft sets named “soft

ω_{δ}

-open sets”. We find that this class constitutes a soft topology that lies strictly between the soft topologies of soft

δ

-open sets and soft

ω^{0}

-open [...] Read more.

In this paper, we present a novel family of soft sets named “soft

ω_{δ}

-open sets”. We find that this class constitutes a soft topology that lies strictly between the soft topologies of soft

δ

-open sets and soft

ω^{0}

-open sets. Also, we introduce certain sufficient conditions for the equivalence between this new soft topology and several existing soft topologies. Moreover, we verify several relationships that contain soft covering properties, such as soft compactness and soft Lindelofness, which are related to this new soft topology. Furthermore, in terms of the soft interior operator in certain soft topologies, we define four classes of soft sets. Via them, we obtain new decomposition theorems for soft

δ

-openness and soft

θ

-openness, and we characterize the soft topological spaces that have the soft “semi-regularization property”. In addition, via soft

ω_{δ}

-open sets, we introduce and investigate a new class of soft functions named “soft

ω_{δ}

-continuous functions”. Finally, we look into the connections between the newly proposed soft concepts and their counterparts in classical topological spaces. Full article

(This article belongs to the Special Issue Advances and Applications of Soft Computing)

24 pages, 5387 KiB

Open AccessArticle

Method for the Statistical Analysis of the Signals Generated by an Acquisition Card for Pulse Measurement

by Yaquelin Verenice Pantoja-Pacheco and Javier Yáñez-Mendiola

Mathematics 2024, 12(6), 923; https://doi.org/10.3390/math12060923 - 21 Mar 2024

Cited by 2 | Viewed by 977

Abstract

This article shows a method for the statistical analysis of signals. Firstly, this method was applied to analyze the processing of signs generated by an acquisition card for pulse measurement using the synchronous demodulation method. The application of the method allowed the study [...] Read more.

This article shows a method for the statistical analysis of signals. Firstly, this method was applied to analyze the processing of signs generated by an acquisition card for pulse measurement using the synchronous demodulation method. The application of the method allowed the study of each signal consisting of a descriptive statistical analysis, followed by the analysis of the trend and dynamics of the movement using the augmented Dickey–Fuller test and Hurst exponent, respectively. Secondarily, the method presented here supported the comparison between the pulse signals obtained by synchronous demodulation and plethysmography methods. In addition, the residuals from the pulse comparison of both methods were analyzed. To quantify the differences between the signals, these were compared using the mean-squared error, the root-mean-square error, the mean absolute error, the mean error, the mean absolute percentage error, and the mean percentage error. After this research, it was possible to analyze the signals knowing characteristics such as the following: the presence of normal, exponential, lognormal, and uniform distributions, stationary trend, and dynamic movement anti-persistent. The novelty that this article proposes is the use of concepts traditionally used in the study of time series and models of demand administration, now focused on supporting improvements over the different stages of design and conceptualization of signal processing devices. Full article

(This article belongs to the Special Issue Data-Driven Statistical Methods)

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14 pages, 457 KiB

Open AccessArticle

Counting Rules for Computing the Number of Independent Sets of a Grid Graph

by Guillermo De Ita Luna, Pedro Bello López and Raymundo Marcial-Romero

Mathematics 2024, 12(6), 922; https://doi.org/10.3390/math12060922 - 21 Mar 2024

Cited by 2 | Viewed by 1128

Abstract

The issue of counting independent sets of a graph, G, represented as

i (G)

, is a significant challenge within combinatorial mathematics. This problem finds practical applications across various fields, including mathematics, computer science, physics, and chemistry. In chemistry, [...] Read more.

The issue of counting independent sets of a graph, G, represented as

i (G)

, is a significant challenge within combinatorial mathematics. This problem finds practical applications across various fields, including mathematics, computer science, physics, and chemistry. In chemistry,

i (G)

is recognized as the Merrifield–Simmons (M-S) index for molecular graphs, which is one of the most relevant topological indices related to the boiling point in chemical compounds. This article introduces an innovative algorithm designed for tallying independent sets within grid-like structures. The proposed algorithm is based on the ‘branch-and-bound’ technique and is applied to compute

i (G_{m, n})

for a square grid formed by m rows and n columns. The proposed approach incorporates the widely recognized vertex reduction rule as the basis for splitting the current subgraph. The methodology involves breaking down the initial grid iteratively until outerplanar graphs are achieved, serving as the ’basic cases’ linked to the leaf nodes of the computation tree or when no neighborhood is incident to a minimum of five rectangular internal faces. The time complexity of the branch-and-bound algorithm speeds up the computation of

i (G_{m, n})

compared to traditional methods, like the transfer matrix method. Furthermore, the scope of the proposed algorithm is more general than the algorithms focused on grids since it could be applied to process general mesh graphs. Full article

(This article belongs to the Special Issue Mathematics and Applications)

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11 pages, 341 KiB

Open AccessArticle

Chi-Square Approximation for the Distribution of Individual Eigenvalues of a Singular Wishart Matrix

by Koki Shimizu and Hiroki Hashiguchi

Mathematics 2024, 12(6), 921; https://doi.org/10.3390/math12060921 - 20 Mar 2024

Viewed by 1103

Abstract

This paper discusses the approximate distributions of eigenvalues of a singular Wishart matrix. We give the approximate joint density of eigenvalues by Laplace approximation for the hypergeometric functions of matrix arguments. Furthermore, we show that the distribution of each eigenvalue can be approximated [...] Read more.

This paper discusses the approximate distributions of eigenvalues of a singular Wishart matrix. We give the approximate joint density of eigenvalues by Laplace approximation for the hypergeometric functions of matrix arguments. Furthermore, we show that the distribution of each eigenvalue can be approximated by the chi-square distribution with varying degrees of freedom when the population eigenvalues are infinitely dispersed. The derived result is applied to testing the equality of eigenvalues in two populations. Full article

(This article belongs to the Special Issue Theory and Applications of Random Matrix)

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17 pages, 514 KiB

Open AccessArticle

Federated Learning with Efficient Aggregation via Markov Decision Process in Edge Networks

by Tongfei Liu, Hui Wang and Maode Ma

Mathematics 2024, 12(6), 920; https://doi.org/10.3390/math12060920 - 20 Mar 2024

Viewed by 1137

Abstract

Federated Learning (FL), as an emerging paradigm in distributed machine learning, has received extensive research attention. However, few works consider the impact of device mobility on the learning efficiency of FL. In fact, it is detrimental to the training result if heterogeneous clients [...] Read more.

Federated Learning (FL), as an emerging paradigm in distributed machine learning, has received extensive research attention. However, few works consider the impact of device mobility on the learning efficiency of FL. In fact, it is detrimental to the training result if heterogeneous clients undergo migration or are in an offline state during the global aggregation process. To address this issue, the Optimal Global Aggregation strategy (OGAs) is proposed. The OGAs first models the interaction between clients and servers of the FL as a Markov Decision Process (MDP) model, which jointly considers device mobility and data heterogeneity to determine local participants that are conducive to global aggregation. To obtain the optimal client participation strategy, an improved

σ

-value iteration method is utilized to solve the MDP, ensuring that the number of participating clients is maintained within an optimal interval in each global round. Furthermore, the Principal Component Analysis (PCA) is used to reduce the dimensionality of the original features to deal with the complex state space in the MDP. The experimental results demonstrate that, compared with other existing aggregation strategies, the OGAs has the faster convergence speed and the higher training accuracy, which significantly improves the learning efficiency of the FL. Full article

(This article belongs to the Special Issue Advances in Mobile Network and Intelligent Communication)

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29 pages, 444 KiB

Open AccessArticle

Generalized Matrix Spectral Factorization with Symmetry and Construction of Quasi-Tight Framelets over Algebraic Number Fields

by Ran Lu

Mathematics 2024, 12(6), 919; https://doi.org/10.3390/math12060919 - 20 Mar 2024

Cited by 1 | Viewed by 739

Abstract

The rational field

Q

is highly desired in many applications. Algorithms using the rational number field

Q

algebraic number fields use only integer arithmetics and are easy to implement. Therefore, studying and designing systems and expansions with coefficients in

Q

or algebraic number [...] Read more.

The rational field

Q

is highly desired in many applications. Algorithms using the rational number field

Q

algebraic number fields use only integer arithmetics and are easy to implement. Therefore, studying and designing systems and expansions with coefficients in

Q

or algebraic number fields is particularly interesting. This paper discusses constructing quasi-tight framelets with symmetry over an algebraic field. Compared to tight framelets, quasi-tight framelets have very similar structures but much more flexibility in construction. Several recent papers have explored the structure of quasi-tight framelets. The construction of symmetric quasi-tight framelets directly applies the generalized spectral factorization of

2 \times 2

matrices of Laurent polynomials with specific symmetry structures. We adequately formulate the latter problem and establish the necessary and sufficient conditions for such a factorization over a general subfield

F

of

C

, including algebraic number fields as particular cases. Our proofs of the main results are constructive and thus serve as a guideline for construction. We provide several examples to demonstrate our main results. Full article

(This article belongs to the Special Issue Matrix Factorization for Signal Processing and Machine Learning)

15 pages, 1189 KiB

Open AccessArticle

Analysis of Shielded Harmonic and Biharmonic Systems by the Iterative Extension Method

by Andrey Ushakov, Sergei Aliukov, Evgeny Meltsaykin and Maksim Eremchuk

Mathematics 2024, 12(6), 918; https://doi.org/10.3390/math12060918 - 20 Mar 2024

Cited by 1 | Viewed by 808

Abstract

To describe stationary physical systems, well-known boundary problems for shielded Poisson and Sophie Germain equations are used. The obtained shielded harmonic and biharmonic systems are approximated using the finite element method and fictitiously continued. The resulting problems are solved using the developed method [...] Read more.

To describe stationary physical systems, well-known boundary problems for shielded Poisson and Sophie Germain equations are used. The obtained shielded harmonic and biharmonic systems are approximated using the finite element method and fictitiously continued. The resulting problems are solved using the developed method of iterative extensions. To expedite the convergence of this method, the relationships between physical quantities on the extension of systems and additional parameters of the iterative method are employed. The formulations of sufficient convergence conditions for the iterative process utilize interdisciplinary connections with functional analysis, applying discrete analogs of the principles of function extensions while preserving norm and class. In the algorithmic implementation of the iterative extensions method, automation is applied to control the selection of the optimal iterative parameter value during information processing. In accordance with the fictitious domain methodology, solvable problems from domains with a complex geometry are reduced to problems in a rectangle in the two-dimensional case and in a rectangular parallelepiped in the three-dimensional case. But now, in the problems being solved, the minimization of the error of the iterative processes is carried out with a norm stronger than the energy norm. Then, all relative errors are estimated from above in the used norms by terms of infinitely decreasing geometric progressions. A generalization of the developed methodology to boundary value problems for polyharmonic equations is possible. Full article

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26 pages, 6306 KiB

Open AccessArticle

Stability and Synchronization of Delayed Quaternion-Valued Neural Networks under Multi-Disturbances

by Jibin Yang, Xiaohui Xu, Quan Xu, Haolin Yang and Mengge Yu

Mathematics 2024, 12(6), 917; https://doi.org/10.3390/math12060917 - 20 Mar 2024

Cited by 1 | Viewed by 851

Abstract

This paper discusses a type of mixed-delay quaternion-valued neural networks (QVNNs) under impulsive and stochastic disturbances. The considered QVNNs model are treated as a whole, rather than as complex-valued neural networks (NNs) or four real-valued NNs. Using the vector Lyapunov function method, some [...] Read more.

This paper discusses a type of mixed-delay quaternion-valued neural networks (QVNNs) under impulsive and stochastic disturbances. The considered QVNNs model are treated as a whole, rather than as complex-valued neural networks (NNs) or four real-valued NNs. Using the vector Lyapunov function method, some criteria are provided for securing the mean-square exponential stability of the mixed-delay QVNNs under impulsive and stochastic disturbances. Furthermore, a type of chaotic QVNNs under stochastic and impulsive disturbances is considered using a previously established stability analysis method. After the completion of designing the linear feedback control law, some sufficient conditions are obtained using the vector Lyapunov function method for determining the mean-square exponential synchronization of drive–response systems. Finally, two examples are provided to demonstrate the correctness and feasibility of the main findings and one example is provided to validate the use of QVNNs for image associative memory. Full article

(This article belongs to the Section Dynamical Systems)

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21 pages, 5185 KiB

Open AccessArticle

Meta-Interpretive LEarning with Reuse

by Rong Wang, Jun Sun, Cong Tian and Zhenhua Duan

Mathematics 2024, 12(6), 916; https://doi.org/10.3390/math12060916 - 20 Mar 2024

Viewed by 974

Abstract

Inductive Logic Programming (ILP) is a research field at the intersection between machine learning and logic programming, focusing on developing a formal framework for inductively learning relational descriptions in the form of logic programs from examples and background knowledge. As an emerging method [...] Read more.

Inductive Logic Programming (ILP) is a research field at the intersection between machine learning and logic programming, focusing on developing a formal framework for inductively learning relational descriptions in the form of logic programs from examples and background knowledge. As an emerging method of ILP, Meta-Interpretive Learning (MIL) leverages the specialization of a set of higher-order metarules to learn logic programs. In MIL, the input includes a set of examples, background knowledge, and a set of metarules, while the output is a logic program. MIL executes a depth-first traversal search, where its program search space expands polynomially with the number of predicates in the provided background knowledge and exponentially with the number of clauses in the program, sometimes even leading to search collapse. To address this challenge, this study introduces a strategy that employs the concept of reuse, specifically through the integration of auxiliary predicates, to reduce the number of clauses in programs and improve the learning efficiency. This approach focuses on the proactive identification and reuse of common program patterns. To operationalize this strategy, we introduce MILER, a novel method integrating a predicate generator, program learner, and program evaluator. MILER leverages frequent subgraph mining techniques to detect common patterns from a limited dataset of training samples, subsequently embedding these patterns as auxiliary predicates into the background knowledge. In our experiments involving two Visual Question Answering (VQA) tasks and one program synthesis task, we assessed MILER’s approach to utilizing reusable program patterns as auxiliary predicates. The results indicate that, by incorporating these patterns, MILER identifies reusable program patterns, reduces program clauses, and directly decreases the likelihood of timeouts compared to traditional MIL. This leads to improved learning success rates by optimizing computational efforts. Full article

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12 pages, 536 KiB

Open AccessArticle

A Divestment Model: Migration to Green Energy Investment Portfolio Concept

by Gaoganwe Sophie Moagi, Obonye Doctor and Edward Lungu

Mathematics 2024, 12(6), 915; https://doi.org/10.3390/math12060915 - 20 Mar 2024

Viewed by 863

Abstract

In a targeted terminal wealth generated by bond and risky assets, where the proportion of a risky asset is gradually being phased down, we propose a divestment model in a risky asset compensated by growth in a bond (insurance). The model includes the [...] Read more.

In a targeted terminal wealth generated by bond and risky assets, where the proportion of a risky asset is gradually being phased down, we propose a divestment model in a risky asset compensated by growth in a bond (insurance). The model includes the phase-down rate of the risky asset,

c (t),

the variable proportion,

π (t),

in a risky asset and the interest rate,

r,

of the bond. To guide the growth of the total wealth in this study, we compared it to the Øksendal and Sulem (Backward Stochastic Differential Equations and Risk Measures (2019)) total wealth for which

c (t) = 0,

and

π (t)

is a constant. We employed the Fokker–Planck equation to find the variable moment,

π (t),

and the associated variance. We proved the existence and uniqueness of the first moment by Feller’s criteria. We have found a pair

(c^{*} (t), r^{*})

for each

π (t)

, which guarantees a growing total wealth. We have addressed the question whether this pair can reasonably be achieved to ensure an acceptable phase-down rate at a financially achievable interest rate,

r^{*}

. Full article

(This article belongs to the Special Issue Financial Mathematics and Applications)

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17 pages, 904 KiB

Open AccessArticle

A Multi-Information Dissemination Model Based on Cellular Automata

by Changheng Shao, Fengjing Shao, Xin Liu, Dawei Yang, Rencheng Sun, Lili Zhang and Kaiwen Jiang

Mathematics 2024, 12(6), 914; https://doi.org/10.3390/math12060914 - 20 Mar 2024

Cited by 2 | Viewed by 1244

Abstract

Significant public opinion events often trigger pronounced fluctuations in online discourse. While existing models have been extensively employed to analyze the propagation of public opinion, they frequently overlook the intricacies of information dissemination among heterogeneous users. To comprehensively address the implications of public [...] Read more.

Significant public opinion events often trigger pronounced fluctuations in online discourse. While existing models have been extensively employed to analyze the propagation of public opinion, they frequently overlook the intricacies of information dissemination among heterogeneous users. To comprehensively address the implications of public opinion outbreaks, it is crucial to accurately predict the evolutionary trajectories of such events, considering the dynamic interplay of multiple information streams. In this study, we propose a SEInR model based on cellular automata to simulate the propagation dynamics of multi-information. By delineating information dissemination rules that govern the diverse modes of information propagation within the network, we achieve precise forecasts of public opinion trends. Through the concurrent simulation and prediction of multi-information game and evolution processes, employing Weibo users as nodes to construct a public opinion cellular automaton, our experimental analysis reveals a significant similarity exceeding 98% between the proposed model and the actual user participation curve observed on the Weibo platform. Full article

(This article belongs to the Special Issue Multi-attribute Decision Making and Intelligent Computing in Smart Governance)

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24 pages, 1378 KiB

Open AccessArticle

Population Feasibility State Guided Autonomous Constrained Multi-Objective Evolutionary Optimization

by Mingcheng Zuo and Yuan Xue

Mathematics 2024, 12(6), 913; https://doi.org/10.3390/math12060913 - 20 Mar 2024

Cited by 2 | Viewed by 923

Abstract

Many practical problems can be classified as constrained multi-objective optimization problems. Although various methods have been proposed for solving constrained multi-objective optimization problems, there is still a lack of research considering the integration of multiple constraint handling techniques. Given this, this paper combines [...] Read more.

Many practical problems can be classified as constrained multi-objective optimization problems. Although various methods have been proposed for solving constrained multi-objective optimization problems, there is still a lack of research considering the integration of multiple constraint handling techniques. Given this, this paper combines the objective and constraint separation method with the multi-operator method, proposing a population feasibility state guided autonomous constrained evolutionary optimization method. This method first defines the feasibility state of the population based on both feasibility and

ε

feasibility of the solutions. Subsequently, a reinforcement learning model is employed to construct a mapping model between the population state and reproduction operators. Finally, based on the real-time population state, the mapping model is utilized to recommend the promising reproduction operator for the next generation. This approach demonstrates significant performance improvement for

ε

constrained mechanisms in constrained multi-objective optimization algorithms, and shows considerable advantages in comparison with state-of-the-art constrained multi-objective optimization algorithms. Full article

(This article belongs to the Section Computational and Applied Mathematics)

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14 pages, 297 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Cohen–Macaulayness of Vertex Splittable Monomial Ideals

by Marilena Crupi and Antonino Ficarra

Mathematics 2024, 12(6), 912; https://doi.org/10.3390/math12060912 - 20 Mar 2024

Viewed by 922

Abstract

In this paper, we give a new criterion for the Cohen–Macaulayness of vertex splittable ideals, a family of monomial ideals recently introduced by Moradi and Khosh-Ahang. Our result relies on a Betti splitting of the ideal and provides an inductive way of checking [...] Read more.

In this paper, we give a new criterion for the Cohen–Macaulayness of vertex splittable ideals, a family of monomial ideals recently introduced by Moradi and Khosh-Ahang. Our result relies on a Betti splitting of the ideal and provides an inductive way of checking the Cohen–Macaulay property. As a result, we obtain characterizations for Gorenstein, level and pseudo-Gorenstein vertex splittable ideals. Furthermore, we provide new and simpler combinatorial proofs of known Cohen–Macaulay criteria for several families of monomial ideals, such as (vector-spread) strongly stable ideals and (componentwise) polymatroidals. Finally, we characterize the family of bi-Cohen–Macaulay graphs by the novel criterion for the Cohen–Macaulayness of vertex splittable ideals. Full article

(This article belongs to the Special Issue Combinatorics and Computation in Commutative Algebra)

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23 pages, 5660 KiB

Open AccessArticle

Profiling Social Sentiment in Times of Health Emergencies with Information from Social Networks and Official Statistics

by Jorge-Eusebio Velasco-López, Ramón-Alberto Carrasco, Jesús Serrano-Guerrero and Francisco Chiclana

Mathematics 2024, 12(6), 911; https://doi.org/10.3390/math12060911 - 20 Mar 2024

Viewed by 1067

Abstract

Social networks and official statistics have become vital sources of information in times of health emergencies. The ability to monitor and profile social sentiment is essential for understanding public perception and response in the context of public health crises, such as the one [...] Read more.

Social networks and official statistics have become vital sources of information in times of health emergencies. The ability to monitor and profile social sentiment is essential for understanding public perception and response in the context of public health crises, such as the one resulting from the COVID-19 pandemic. This study will explore how social sentiment monitoring and profiling can be conducted using information from social networks and official statistics, and how this combination of data can offer a more complete picture of social dynamics in times of emergency, providing a valuable tool for understanding public perception and guiding a public health response. To this end, a three-layer architecture based on Big Data and Artificial Intelligence is presented: the first layer focuses mainly on collecting, storing, and governing the necessary data such as social media and official statistics; in the second layer, the representation models and machine learning necessary for knowledge generation are built, and in the third layer the previously generated knowledge is adapted for better understanding by crisis managers through visualization techniques among others. Based on this architecture, a KDD (Knowledge Discovery in Databases) framework is implemented using methodological tools such as sentiment analysis, fuzzy 2-tuple linguistic models and time series prediction with the Prophet model. As a practical demonstration of the proposed model, we use tweets as data source (from the social network X, formerly known as Twitter) generated during the COVID-19 pandemic lockdown period in Spain, which are processed to identify the overall sentiment using sentiment analysis techniques and fuzzy linguistic variables, and combined with official statistical indicators for prediction, visualizing the results through dashboards. Full article

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11 pages, 235 KiB

Open AccessArticle

Property (A) and Oscillation of Higher-Order Trinomial Differential Equations with Retarded and Advanced Arguments

by Blanka Baculikova

Mathematics 2024, 12(6), 910; https://doi.org/10.3390/math12060910 - 20 Mar 2024

Viewed by 705

Abstract

In this paper, a new effective technique for the investigation of the higher-order trinomial differential equations [...] Read more.

In this paper, a new effective technique for the investigation of the higher-order trinomial differential equations

y^{(n)} (t) + p (t) y (τ (t)) + q (t) y (σ (t)) = 0

is established. We offer new criteria for so-called property (A) and oscillation of the considered equation. Examples are provided to illustrate the importance of our results. Full article

(This article belongs to the Section Difference and Differential Equations)

22 pages, 718 KiB

Open AccessArticle

A Forward–Backward Simheuristic for the Stochastic Capacitated Dispersion Problem

by Juan F. Gomez, Anna Martínez-Gavara, Javier Panadero, Angel A. Juan and Rafael Martí

Mathematics 2024, 12(6), 909; https://doi.org/10.3390/math12060909 - 20 Mar 2024

Cited by 1 | Viewed by 1031

Abstract

In an effort to balance the distribution of services across a given territory, dispersion and diversity models typically aim to maximize the minimum distance between any pair of facilities. Specifically, in the capacitated dispersion problem (CDP), each facility has an associated capacity or [...] Read more.

In an effort to balance the distribution of services across a given territory, dispersion and diversity models typically aim to maximize the minimum distance between any pair of facilities. Specifically, in the capacitated dispersion problem (CDP), each facility has an associated capacity or level of service, and the objective is to select a set of facilities so that the minimum distance between any pair of them (dispersion) is maximized, while ensuring a user-defined level of service. This problem can be formulated as a linear integer model, where the sum of the capacities of the selected facilities must match or exceed the total demand in the network. Real-life applications often necessitate considering the levels of uncertainty affecting the capacity of the nodes. Failure to account for this uncertainty could lead to low-quality or infeasible solutions in practical scenarios. However, research addressing the stochastic version of the CDP is scarce. This paper introduces two models for the CDP with stochastic capacities, incorporating soft constraints and penalty costs for violating the total capacity constraint. The first model includes a probabilistic constraint to ensure the required level of service with a certain probability, while the second model introduces a soft constraint with penalty costs for violations. To solve both variants of the model, a forward–backward simheuristic algorithm is proposed. Our approach combines a metaheuristic algorithm with Monte Carlo simulation, enabling the efficient handling of the random behavior of node capacities and obtaining reliable solutions regardless of their probability distribution. Full article

(This article belongs to the Special Issue Operations Research and Intelligent Computing for System Optimization)

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23 pages, 2123 KiB

Open AccessArticle

Statistical Analysis of the Negative–Positive Transformation in Image Encryption

by Manuel Alejandro Cardona-López, Juan Carlos Chimal-Eguía, Víctor Manuel Silva-García and Rolando Flores-Carapia

Mathematics 2024, 12(6), 908; https://doi.org/10.3390/math12060908 - 20 Mar 2024

Viewed by 983

Abstract

The negative–positive transformation (NPT) is a widely employed technique for encrypting images on pixel blocks, commonly integrated into cryptosystems compatible with compression algorithms. The existing literature on NPT analysis can be categorized into two types: theoretical analyses with results that apply to any [...] Read more.

The negative–positive transformation (NPT) is a widely employed technique for encrypting images on pixel blocks, commonly integrated into cryptosystems compatible with compression algorithms. The existing literature on NPT analysis can be categorized into two types: theoretical analyses with results that apply to any image, primarily focused on compression compatibility, and numerical analyses that report empirical results from specific images, some without explaining the causes of the security results, while others are only related to the compression performance. Consequently, there is a significant gap in understanding the implications of applying the NPT for data protection. For that reason, this paper conducts a theoretical statistical analysis, presenting, demonstrating, and verifying six theorems to understand the security contributions of NPT. Two theorems examine the shape of the image histogram and the scatter plot of adjacent pixels after the NPT application. The subsequent four theorems explore the influence of NPT on the mean, variance, covariance, and correlation within each pixel block. The findings indicate that the NPT generates images with symmetrical histograms, the correlation of pixel blocks remains invariant, and distinct vertical and horizontal reflections manifest on the scatter plot. These theorems are verified by encrypting the Lena image with four pixel-block sizes. The histogram symmetry passed the goodness-of-fit test at a significance level of 5%, revealing consistent results. The correlation of pixel blocks remained unchanged, and the scatter plot exhibited an x-shaped pattern. Therefore, as the NPT alone does not achieve desirable encryption results, such as uniform histograms, scatter plots, and decreasing correlation, cryptosystems should complement it with additional techniques. Full article

(This article belongs to the Special Issue Computational Algebra, Coding Theory and Cryptography)

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17 pages, 4865 KiB

Open AccessArticle

The Influence of Platen Stiffness on a Specimen’s Failure Initiation Point and the Failure Pattern of Brittle Materials in the Standardized Uniaxial Compression Test

by Diego-José Guerrero-Miguel, Martina-Inmaculada Álvarez-Fernández, Ramón Gutiérrez-Moizant, María-Belén Prendes-Gero and Celestino González-Nicieza

Mathematics 2024, 12(6), 907; https://doi.org/10.3390/math12060907 - 20 Mar 2024

Cited by 1 | Viewed by 961

Abstract

Uniaxial compressive strength is a variable necessary for adequately characterizing a material’s mechanical properties. However, a specimen’s geometric deviations and elastic properties may lead to undesirable stress states, which cause strong discrepancies between the results of the uniaxial compression test and its theoretical [...] Read more.

Uniaxial compressive strength is a variable necessary for adequately characterizing a material’s mechanical properties. However, a specimen’s geometric deviations and elastic properties may lead to undesirable stress states, which cause strong discrepancies between the results of the uniaxial compression test and its theoretical foundations. While geometric deviations may cause non-uniform contact between the platen and the specimen, elastic properties can provoke severe end effects that disturb the local stress field near the points of contact. To address how the relative stiffness between the platen and the specimen influences the induced stress field, numerical simulations considering the stiffness ratios

E_{p} / E_{s} = 3

,

E_{p} / E_{s} = 1

and

E_{p} / E_{s} = 0.05

were performed. Subsequently, these results were employed to establish the relation between relative stiffness and specimen failure patterns in brittle materials, particularly in three different rocks. The results prove that the platen stiffness must be accurately selected to match that of the tested material, in order to avoid undesirable local stress fields near the point of contact and to induce homogeneous uniaxial compression that guarantees reliable uniaxial compressive strength characterization. Furthermore, the brittle failure patterns reported in previous studies were correlated with the induced stress fields inside the specimen depending on its platen stiffness, allowing the validity of the test results to be verified based on a simple visual inspection. Full article

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19 pages, 2858 KiB

Open AccessArticle

Queuing-Inventory System with Catastrophes in the Warehouse: Case of Rare Catastrophes

by Agassi Melikov, Laman Poladova and Janos Sztrik

Mathematics 2024, 12(6), 906; https://doi.org/10.3390/math12060906 - 19 Mar 2024

Viewed by 822

Abstract

A model of a single-server queuing-inventory system (QIS) with a limited waiting buffer for consumer customers (c-customers) and catastrophes has been developed. When a catastrophe occurs, all items in the system’s warehouse are destroyed, but c-customers in the system are [...] Read more.

A model of a single-server queuing-inventory system (QIS) with a limited waiting buffer for consumer customers (c-customers) and catastrophes has been developed. When a catastrophe occurs, all items in the system’s warehouse are destroyed, but c-customers in the system are still waiting for replenishment. In addition to c-customers, negative customers (n-customers) are also taken into account, each of which displaces one c-customer (if any). The policy (s, S) is used to replenish stocks. If, when a customer enters, the system warehouse is empty, then, according to Bernoulli’s trials, this customer either leaves the system without goods or joins the buffer. The mathematical model of the investigated QIS is constructed in the form of a continuous-time Markov chain (CTMC). Both exact and approximate methods for calculating the steady-state probabilities of constructed CTMCs are proposed and closed-form expressions are obtained for calculating the performance measures. Numerical evaluations are presented, demonstrating the high accuracy of the developed approximate formulas, as well as the behavior of performance measures depending on the input parameters. In addition, an optimization problem is solved to obtain the optimal value of the reorder point to minimize the expected total cost. Full article

(This article belongs to the Special Issue Advances in Queueing Theory, 2nd Edition)

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28 pages, 409 KiB

Open AccessArticle

5th-Order Multivariate Edgeworth Expansions for Parametric Estimates

by C. S. Withers

Mathematics 2024, 12(6), 905; https://doi.org/10.3390/math12060905 - 19 Mar 2024

Viewed by 778

Abstract

The only cases where exact distributions of estimates are known is for samples from exponential families, and then only for special functions of the parameters. So statistical inference was traditionally based on the asymptotic normality of estimates. To improve on this we need [...] Read more.

The only cases where exact distributions of estimates are known is for samples from exponential families, and then only for special functions of the parameters. So statistical inference was traditionally based on the asymptotic normality of estimates. To improve on this we need the Edgeworth expansion for the distribution of the standardised estimate. This is an expansion in

n^{- 1 / 2}

about the normal distribution, where n is typically the sample size. The first few terms of this expansion were originally given for the special case of a sample mean. In earlier work we derived it for any standard estimate, hugely expanding its application. We define an estimate

\hat{w}

of an unknown vector w in

R^{p}

, as a standard estimate, if

E \hat{w} \to w

as

n \to \infty

, and for

r \geq 1

the rth-order cumulants of

\hat{w}

have magnitude

n^{1 - r}

and can be expanded in

n^{- 1} .

Here we present a significant extension. We give the expansion of the distribution of any smooth function of

\hat{w}

, say

t (\hat{w})

in

R^{q},

giving its distribution to

n^{- 5 / 2}

. We do this by showing that

t (\hat{w})

, is a standard estimate of

t (w)

. This provides far more accurate approximations for the distribution of

t (\hat{w})

than its asymptotic normality. Full article

(This article belongs to the Special Issue Advances in Applied Probability and Statistical Inference)

31 pages, 2716 KiB

Open AccessArticle

Uncertain Asymptotic Stability Analysis of a Fractional-Order System with Numerical Aspects

by Safoura Rezaei Aderyani, Reza Saadati, Donal O’Regan and Fehaid Salem Alshammari

Mathematics 2024, 12(6), 904; https://doi.org/10.3390/math12060904 - 19 Mar 2024

Viewed by 818

Abstract

We apply known special functions from the literature (and these include the Fox

H -

function, the exponential function, the Mittag-Leffler function, the Gauss Hypergeometric function, the Wright function, the

G -

function, the Fox–Wright function and the Meijer

G -

function) and [...] Read more.

We apply known special functions from the literature (and these include the Fox

H -

function, the exponential function, the Mittag-Leffler function, the Gauss Hypergeometric function, the Wright function, the

G -

function, the Fox–Wright function and the Meijer

G -

function) and fuzzy sets and distributions to construct a new class of control functions to consider a novel notion of stability to a fractional-order system and the qualified approximation of its solution. This new concept of stability facilitates the obtention of diverse approximations based on the various special functions that are initially chosen and also allows us to investigate maximal stability, so, as a result, enables us to obtain an optimal solution. In particular, in this paper, we use different tools and methods like the Gronwall inequality, the Laplace transform, the approximations of the Mittag-Leffler functions, delayed trigonometric matrices, the alternative fixed point method, and the variation of constants method to establish our results and theory. Full article

(This article belongs to the Special Issue Fractional Calculus: Advances and Applications)

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15 pages, 2094 KiB

Open AccessArticle

Optimal Design of Group Orthogonal Phase-Coded Waveforms for MIMO Radar

by Tianqu Liu, Jinping Sun, Guohua Wang, Xianxun Yao and Yaqiong Qiao

Mathematics 2024, 12(6), 903; https://doi.org/10.3390/math12060903 - 19 Mar 2024

Viewed by 1175

Abstract

Digital radio frequency memory (DRFM) has emerged as an advanced technique to achieve a range of jamming signals, due to its capability to intercept waveforms within a short time. multiple-input multiple-output (MIMO) radars can transmit agile orthogonal waveform sets for different pulses to [...] Read more.

Digital radio frequency memory (DRFM) has emerged as an advanced technique to achieve a range of jamming signals, due to its capability to intercept waveforms within a short time. multiple-input multiple-output (MIMO) radars can transmit agile orthogonal waveform sets for different pulses to combat DRFM-based jamming, where any two groups of waveform sets are also orthogonal. In this article, a group orthogonal waveform optimal design model is formulated in order to combat DRFM-based jamming by flexibly designing waveforms for MIMO radars. Aiming at balancing the intra- and intergroup orthogonal performances, the objective function is defined as the weighted sum of the intra- and intergroup orthogonal performance metrics. To solve the formulated model, in this article, a group orthogonal waveform design algorithm is proposed. Based on a primal-dual-type method and proper relaxations, the proposed algorithm transforms the original problem into a series of simple subproblems. Numerical results demonstrate that the obtained group orthogonal waveforms have the ability to flexibly suppress DRFM-based deceptive jamming, which is not achievable using p-majorization–minimization (p-MM) and primal-dual, two of the most advanced orthogonal waveform design algorithms. Full article

(This article belongs to the Special Issue Advanced Algorithms in Wireless Communication and Internet of Things (IoT))

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13 pages, 3929 KiB

Open AccessArticle

Inter-Reconfigurable Robot Path Planner for Double-Pass Complete Coverage Problem

by Ash Wan Yaw Sang, Zhenyuan Yang, Lim Yi, Chee Gen Moo, Rajesh Elara Mohan and Anh Vu Le

Mathematics 2024, 12(6), 902; https://doi.org/10.3390/math12060902 - 19 Mar 2024

Viewed by 1212

Abstract

Recent advancements in autonomous mobile robots have led to significant progress in area coverage tasks. However, challenges persist in optimizing the efficiency and computational complexity of complete coverage path planner (CCPP) algorithms for multi-robot systems, particularly in scenarios requiring revisiting or a double [...] Read more.

Recent advancements in autonomous mobile robots have led to significant progress in area coverage tasks. However, challenges persist in optimizing the efficiency and computational complexity of complete coverage path planner (CCPP) algorithms for multi-robot systems, particularly in scenarios requiring revisiting or a double pass in specific locations, such as cleaning robots addressing spilled consumables. This paper presents an innovative approach to tackling the double-pass complete coverage problem using an autonomous inter-reconfigurable robot path planner. Our solution leverages a modified Glasius bio-inspired neural network (GBNN) to facilitate double-pass coverage through inter-reconfiguration between two robots. We compare our proposed algorithm with traditional multi-robot path planning in a centralized system, demonstrating a reduction in algorithm iterations and computation time. Our experimental results underscore the efficacy of the proposed solution in enhancing the efficiency of area coverage tasks. Furthermore, we discuss the implementation details and limitations of our study, providing insights for future research directions in autonomous robotics. Full article

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22 pages, 5397 KiB

Open AccessArticle

Challenges and Countermeasures of Federated Learning Data Poisoning Attack Situation Prediction

by Jianping Wu, Jiahe Jin and Chunming Wu

Mathematics 2024, 12(6), 901; https://doi.org/10.3390/math12060901 - 19 Mar 2024

Viewed by 2207

Abstract

Federated learning is a distributed learning method used to solve data silos and privacy protection in machine learning, aiming to train global models together via multiple clients without sharing data. However, federated learning itself introduces certain security threats, which pose significant challenges in [...] Read more.

Federated learning is a distributed learning method used to solve data silos and privacy protection in machine learning, aiming to train global models together via multiple clients without sharing data. However, federated learning itself introduces certain security threats, which pose significant challenges in its practical applications. This article focuses on the common security risks of data poisoning during the training phase of federated learning clients. First, the definition of federated learning, attack types, data poisoning methods, privacy protection technology and data security situational awareness are summarized. Secondly, the system architecture fragility, communication efficiency shortcomings, computing resource consumption and situation prediction robustness of federated learning are analyzed, and related issues that affect the detection of data poisoning attacks are pointed out. Thirdly, a review is provided from the aspects of building a trusted federation, optimizing communication efficiency, improving computing power technology and personalized the federation. Finally, the research hotspots of the federated learning data poisoning attack situation prediction are prospected. Full article

(This article belongs to the Special Issue Advanced Analyses and Algorithms for Trustworthy AI Systems and Applications)

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19 pages, 716 KiB

Open AccessArticle

Optimizing Propellant Distribution for Interorbital Transfers

by J. De Curtò and I. De Zarzà

Mathematics 2024, 12(6), 900; https://doi.org/10.3390/math12060900 - 19 Mar 2024

Viewed by 1027

Abstract

The advent of space exploration missions, especially those aimed at establishing a sustainable presence on the Moon and beyond, necessitates the development of efficient propulsion and mission planning techniques. This study presents a comprehensive analysis of chemical and electric propulsion systems for spacecraft, [...] Read more.

The advent of space exploration missions, especially those aimed at establishing a sustainable presence on the Moon and beyond, necessitates the development of efficient propulsion and mission planning techniques. This study presents a comprehensive analysis of chemical and electric propulsion systems for spacecraft, focusing on optimizing propellant distribution for missions involving transfers from Low-Earth Orbit (LEO) to Geostationary Orbit (GEO) and the Lunar surface. Using mathematical modeling and optimization algorithms, we calculate the delta-v requirements for key mission segments and determine the propellant mass required for each propulsion method. The results highlight the trade-offs between the high thrust of chemical propulsion and the high specific impulse of electric propulsion. An optimization model is developed to minimize the total propellant mass, considering a hybrid approach that leverages the advantages of both propulsion types. This research contributes to the field of aerospace engineering by providing insights into propulsion system selection and mission planning for future exploration missions to the Moon, Mars, and Venus. Full article

(This article belongs to the Special Issue Modeling and Simulation in Engineering, 3rd Edition)

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