Mathematical Framework Research Articles

Investigation of Fractional Order Tumor Cell Concentration Equation Using Finite Difference Method

المعادلة التفاضلية الكسرية توفر إطارًا رياضيًا قويًا لنمذجة وفهم مجموعة واسعة من الظواهر المعقدة وغير المحلية والمعتمدة على الذاكرة في مختلف التطبيقات العلمية والهندسية والعالمية. تعد الاعتلالات الكبدية أورامًا سرطانية ثانوية تتطور في الكبد نتيجة انتشار خلايا السرطان منورام سرطاني أصلي ينشأ في جزء آخر من جسم الإنسان. التركيز الأساسي لهذه الدراسة هو فهم نمو الأورام في الكبد البشري، سواء بوجود أو بدون علاج دوائي. ولتحقيق ذلك، يتم استخدام معادلة تفاضلية جزئية معرفة بالترتيب الزمني والكسري، ويتم تحليلها باستخدام أساليب عددية. يتم استخدام المشتقة كابوتو لاستكشاف تأثير العلاج الدوائي على نمو الورم. لحل النموذج الرياضي عدديًا، تم تطوير طريقة الفروق المحدودة كرانك-نيكولسون. يتم اختيار هذه الطريقة بسبب خصائصها المميزة، بما في ذلك الاستقرار الغير مشروط والدقة من الدرجة الثانية في الأبعاد الزمنية والمكانية. الاستقرار والدقة التي توفرهما هاتان الخصائص أمران حاسمان لضمان موثوقية النتائج المحصلة في هذه البحث. يتم تقديم النتائج والتحققات المستمدة من هذه الدراسة بفعالية من خلال تمثيلات بصرية متنوعة. تعتبر هذه الوسائل البصرية أدوات لا غنى عنها لفهم التأثير العميق للعلاج الدوائي على نمو الأورام. من خلال هذا التمثيل البصري، يمكن للشخص الحصول على فهم أوضح وأكثر تفصيلًا للديناميات المعقدة التي تحدث. تم الوصول إلى الحل العددي لهذه المشكلة المعقدة من خلال تنفيذ خوارزميات تم تطويرها بدقة باستخدام لغة البرمجة بايثون المتعددة الاستخدامات والقوية. مرونة بايثون والمكتبات الشاملة وإمكانيات الحساب العددي القوية تجعلها الخيار المثالي للتعامل مع تعقيدات هذه الدراسة.

Advanced Computational Methods for News Classification: A Study in Neural Networks and CNN integrated with GPT

In an era inundated with vast amounts of information, the imperative for efficient news classification is paramount. This research explores the sophisticated integration of neural networks and convolutional neural networks (CNN) with Generative Pre-trained Transformers (GPT) to enhance the precision and efficacy of news categorization. The rapid digital dissemination of news necessitates advanced computational methodologies capable of accurate classification and event prediction that include finance and economic events. Leveraging recent advancements in machine learning and natural language processing (NLP), this study utilizes large language models (LLMs) such as GPT and BERT, known for their exceptional comprehension and generation of human-like text. Over 232 days, our methodology classified 33,979 news articles into Education & Learning, Health & Medicine, and Science & Technology, with further subcategorization into 32 distinct subcategories. For evaluation, a sample of 5,000 articles was assessed using metrics such as True Positive (TP), True Negative (TN), False Positive (FP), False Negative (FN), Precision, Recall, and F1-Score. In comparison with the existing studies, the proposed method achieving significantly higher with average scores of 0.986 (Precision), 0.987 (Recall), and 0.987 (F1-Score). This research offers substantial practical contributions, providing detailed insights into news source contributions, effective anomaly detection, and predictive trend analysis using neural networks. The theoretical contributions are profound, demonstrating the mathematical integration of GPT with CNNs and recurrent neural networks. This integration advances computational news classification and exemplifies how sophisticated mathematical frameworks enhance large-scale text data analysis, marking a pivotal advancement in applying advanced computational methods in real-world scenarios.

Modern chemical graph theory

AbstractGraph theory has a long history in chemistry. Yet as the breadth and variety of chemical data is rapidly changing, so too do graph encoding methods and analyses that yield qualitative and quantitative insights. Using illustrative cases within a basic mathematical framework, we showcase modern chemical graph theory's utility in Chemists' analysis and model development toolkit. The encoding of both experimental and simulation data is discussed at various levels of granularity of information. This is followed by a discussion of the two major classes of graph theoretical analyses: identifying connectivity patterns and partitioning methods. Measures, metrics, descriptors, and topological indices are then introduced with an emphasis upon enhancing interpretability and incorporation into physical models. Challenging data cases are described that include strategies for studying time dependence. Throughout, we incorporate recent advancements in computer science and applied mathematics that are propelling chemical graph theory into new domains of chemical study.This article is categorized under: Molecular and Statistical Mechanics > Molecular Dynamics and Monte‐Carlo Methods Structure and Mechanism > Computational Materials Science Structure and Mechanism > Molecular Structures

Analysis of Solutions for Nonlinear ψ-Caputo Fractional Differential Equations with Fractional Derivative Boundary Conditions in Banach Algebra

This article explores the solutions of nonlinear implicit ψ-Caputo fractional-order ordinary differential equations (NLIFDEs) with two-point fractional derivatives boundary conditions in Banach algebra. The research aims to establish the existence and uniqueness of solutions for this complex class of differential equations. Utilizing Banach’s and Krasnoselskii’s fixed point theorems, the study conducts a rigorous analysis of the solutions, ensuring their existence and uniqueness. This comprehensive investigation contributes to enhancing the understanding of the behavior of solutions of nonlinear fractional differentials within a challenging mathematical framework.

Degenerate Cahn–Hilliard systems: From nonlocal to local

We provide a rigorous mathematical framework to establish the limit of a nonlocal model of cell–cell adhesion system to a local model. When the parameter of the nonlocality goes to 0, the system tends to a Cahn–Hilliard system with degenerate mobility and cross-interaction forces. Our analysis relies on a-priori estimates and compactness properties.

Node Role Selection and Rotation Scheme for Energy Efficiency in Multi-Level IoT-Based Heterogeneous Wireless Sensor Networks (HWSNs).

The emergence of Internet of Things (IoT)-based heterogeneous wireless sensor network (HWSN) technology has become widespread, playing a significant role in the development of diverse human-centric applications. The role of efficient resource utilisation, particularly energy, becomes further critical in IoT-based HWSNs than it was in WSNs. Researchers have proposed numerous approaches to either increase the provisioned resources on network devices or to achieve efficient utilisation of these resources during network operations. The application of a vast proportion of such methods is either limited to homogeneous networks or to a single parameter and limited-level heterogeneity. In this work, we propose a multi-parameter and multi-level heterogeneity model along with a cluster-head rotation method that balances energy and maximizes lifetime. This method achieves up to a 57% increase in throughput to the base station, owing to improved intra-cluster communication in the IoT-based HWSN. Furthermore, for inter-cluster communication, a mathematical framework is proposed that first assesses whether the single-hop or multi-hop inter-cluster communication is more energy efficient, and then computes the region where the next energy-efficient hop should occur. Finally, a relay-role rotation method is proposed among the potential next-hop nodes. Results confirm that the proposed methods achieve 57.44%, 51.75%, and 17.63% increase in throughput of the IoT-based HWSN as compared to RLEACH, CRPFCM, and EERPMS, respectively.

A stochastic averaging mathematical framework for design and optimization of nonlinear energy harvesters with several electrical DOFs

Energy harvesters for mechanical vibrations are electro-mechanical systems designed to capture ambient dispersed kinetic energy, and to convert it into usable electrical power. The random nature of mechanical vibrations, combined with the intrinsic non-linearity of the harvester, implies that long, time domain Monte-Carlo simulations are required to assess the device performance, making the analysis burdensome when a large parameter space must be explored. Therefore a simplified, albeit approximate, semi-analytical analysis technique is of paramount importance. In this work we present a methodology for the analysis and design of nonlinear piezoelectric energy harvesters for random mechanical vibrations. The methodology is based on the combined application of model order reduction, to project the dynamics onto a lower dimensional space, and of stochastic averaging, to calculate the stationary probability density function of the reduced variables. The probability distribution is used to calculate expectations of the most relevant quantities, like output voltage, harvested power and power efficiency. Based on our previous works, we consider an energy harvester with a matching network, interposed between the harvester and the load, that reduces the impedance mismatch between the two stages. The methodology is applied to the optimization of the matching network, allowing to maximize the global harvested power and the conversion efficiency. We show that the proposed methodology gives accurate predictions of the harvester’s performance, and that it can be used to significantly simplify the analysis, design and optimization of the device.

FORMATION OF A SYSTEM OF GOALS FOR MANAGING THE INNOVATION ACTIVITIES OF AN ENTERPRISE IN THE CONDITIONS OF POST-CONFLICT TRANSFORMATION

The article examines the role of innovation in the development of enterprises. An approach to innovation management at an enterprise in the context of post-conflict transformation is developed, based on the premise that the goals of an enterprise should determine the goals of innovation management, and the processes of innovation management at an enterprise should be integrated with the vertical and horizontal levels of economic and production business processes in the context of post-conflict transformation. Based on the results of the study, the mathematical basis for innovation management in the system of methodological approaches (transfer management, algorithmic management, creative management) based on the identified processes of innovation management at enterprises in the context of post-conflict transformation is determined. An analysis of approaches to the formation of goals for managing innovations at enterprises in the context of post-conflict transformation is carried out. The approach, which is contained within 8 key spaces, with the allocation of 5 groups of goals (financial, production, marketing, human resource management, information and technical support) is substantiated. A system of goals for managing the innovation and investment activities of enterprise is formed. The author details managerial actions/decisions at the strategic, tactical and operational levels and specifies tactical functional goals in the context of financial, production, marketing, human resource management and information technology, which ensures further optimization of the use of various types of resources (primarily human resources) at the enterprise. A mathematical framework has been developed for the subsystem at the strategic, tactical and operational levels. The obtained results can be implemented in the activities of enterprises and used in further scientific research.

Stokes Phenomena in Lensing

As lensing of coherent astrophysical sources, e.g., pulsars, fast radio bursts, and gravitational waves, becomes observationally relevant, the mathematical framework of Picard–Lefschetz theory has recently been introduced to fully account for wave optics effects. Accordingly, the concept of lensing images has been generalized to include complex solutions of the lens equation referred to as “imaginary images,” and more radically, to the Lefschetz thimbles, which are a sum of the steepest descent contours connecting the real and imaginary images in the complex domain. In this wave-optics-based theoretical framework of lensing, we study the “Stokes phenomena” as the change of the topology of the Lefschetz thimbles. Similar to the well-known caustics at which the number of geometric images changes abruptly, the corresponding Stokes lines are the boundaries in the parameter space where the number of effective imaginary images changes. We map the Stokes lines for a few lens models. The resulting Stokes line-caustics network represents a unique feature of the lens models. The observable signature of the Stokes phenomena is the change of interference behavior, in particular the onset of frequency oscillation for some Stokes lines. We also demonstrate high-order Stokes phenomena where the system has a continuous number of effective images but with an abrupt change in the way they are connected to each other by the Lefschetz thimbles. Their full characterization calls for an analogy of the catastrophe theory for caustics.

Navigating ambiguity: A novel neutrosophic cubic shapley normalized weighted Bonferroni Mean aggregation operator with application in the investment environment

The Neutrosophic Cubic Shapley Normalized Bonferroni (NC-SNWBM) method represents a cutting-edge approach to decision making theory, combining three distinct mathematical frameworks the neutrosophic cubic sets (NCS), Shapley values, and the Bonferroni aggregation operator. This innovative method addresses the challenges posed by uncertainty, vagueness, and imprecision in decision-making (DM) processes, offering a comprehensive and versatile tool for handling complex and dynamic scenarios. Neutrosophic cubic sets offers a strong platform to handle ambiguous and vague data due to three components Membership Grade (MG), Non-Membership Grade (NMG) and Indeterminancy Grade (IG) in data. By adding Shapley Fuzzy Measures (SFM), which come from cooperative game theory, distribute values among cooperative agents equally and to account for each agent's contributions to all potential coalitions. The Bonferroni aggregation operator—a statistical aggregative tool that regulates the likelihood of many types in error in statistical tests and the interdependence of the input arguments by allowing different values to parameters involved. These values are further improved by normalization in the framework of the NC-SNWBM approach in order to consider the various degrees of impact that agents exert in various circumstances. This operator is smoothly combined with normalized Shapley values and neutrosophic cubic sets in the NC-SNWBM approach to enable the aggregation of data with different levels of imprecision and uncertainty from various sources using NCS. The MG, NMG and IG connected to NCS are important elements of the NC-SNWBM approach. To evaluate each element's contribution to the overall value distribution SFM are used, and the Bonferroni aggregation operator maintains a careful balance between conservatism and significance. Together, these components provide a thorough framework that successfully tackles the problems caused by ambiguity, imprecision, and uncertainty in scenarios involving decision-making. The NC-SNWBM operator is applied to a numerical problem as an application in investment environment and sensitive and comparative analysis are conducted. The recommendation based on sensitive and comparative analysis proposed.

Positivity-Preserving Rational Cubic Fractal Interpolation Function Together with Its Zipper Form

In this paper, a novel class of rational cubic fractal interpolation function (RCFIF) has been proposed, which is characterized by one shape parameter and a linear denominator. In interpolation for shape preservation, the proposed rational cubic fractal interpolation function provides a simple but effective approach. The nature of shape preservation of the proposed rational cubic fractal interpolation function makes them valuable in the field of data visualization, as it is crucial to maintain the original data shape in data visualization. Furthermore, we discussed the upper bound of error and explored the mathematical framework to ensure the convergence of RCFIF. Shape parameters and scaling factors are constraints to obtain the desired shape-preserving properties. We further generalized the proposed RCFIF by introducing the concept of signature, giving its construction in the form of a zipper-rational cubic fractal interpolation function (ZRCFIF). The positivity conditions for the rational cubic fractal interpolation function and zipper-rational cubic fractal interpolation function are found, which required a detailed analysis of the conditions where constraints on shape parameters and scaling factor lead to the desired shape-preserving properties. In the field of shape preservation, the proposed rational cubic fractal interpolation function and zipper fractal interpolation function both represent significant advancement by offering a strong tool for data visualization.

Analytic evolution for complex coupled tight-binding models: Applications to quantum light manipulation

We present analytic solutions to the evolution in generalized tight-binding models, which consider complex first-neighbor couplings with equal amplitude and arbitrary phases. Our findings provide a powerful tool to efficiently calculate expectation values and correlations within the system, which are otherwise difficult to compute numerically. We apply our results to relevant examples in quantum light manipulation using -port linear couplers, describing the evolution of single (multi)-mode squeezing, single-photon added (subtracted) Gaussian states, and second-order site-to-site photon correlations. Significantly, our analytic results outperform standard numerical calculations. Our study paves the way for a comprehensive mathematical framework describing the spatial evolution of quantum states across a wide range of physical systems governed by the tight-binding model. Published by the American Physical Society 2024

Analysis of symbiotic backscatter empowered wireless sensors network with short-packet communications.

Recent progress studies in light of wireless communication systems mainly centred around two focuses: zero-energy consumption and ultra-reliable and low-latency communication (URLLC). Among various cutting-edge areas, exploiting ambient backscatter communication (Backcom) has recently been devised as one of the foremost solutions for achieving zero energy consumption through the viability of ambient radio frequency. Meanwhile, using short-packet communication (SPC) is the cheapest way to reach the goal of URLLCs. Upon these benefits, we investigate the feasibility of Backcom and SPC for symbiotic wireless sensor networks by analyzing the system performance. Specifically, we provide a highly approximated mathematical framework for evaluating the block-error rate (BLER) performance, followed by some useful asymptotic results. These results provide insights into the level of diversity and coding gain, as well as how packet design impacts BLER performance. Numerical results confirm the efficacy of the developed framework and the correctness of key insights gleaned from the asymptotic analyses.

Structure of higher-order interactions in social-ecological networks through Q-analysis of their neighbourhood and clique complex.

This paper studies higher-order interactions in social-ecological networks, which formally represent interactions within the social and ecological units of an ecosystem. Many real-world social ecosystems exhibit not only pairwise interactions but also higher-order interactions among their units. Therefore, the conventional graph-theoretic description of networks falls short of capturing these higher-order interactions due to the inherent limitations of the graph definition. In this work, a mathematical framework for capturing the higher-order interactions of a social-ecological system has been given by incorporating notions from combinatorial algebraic topology. In order to achieve this, two different simplicial complexes, the clique and the neighbourhood complex, have been constructed from a pairwise social-ecological network. As a case study, the Q-analysis and a structural study of the interactions in the rural agricultural system of southern Madagascar have been done at various structural levels denoted by q. The results obtained by calculating all the structural vectors for both simplicial complexes, along with exciting results about the participation of facets of the clique complex at different q-levels, have been discussed. This work also establishes significant theorems concerning the dimension of the neighbourhood complex and clique complex obtained from the parent pairwise network.

A New Second-order Maximum-principle-preserving Finite-volume Method for Flow Problems Involving Discontinuous Coefficients

A new development of Finite Volumes (FV, for short) and its theoretical analysis are the purpose of this work. Recall that FV are known as powerful tools to address equations of conservation laws (mass, energy, momentum,...). Over the last two decades investigators have succeeded in putting in place a mathematical framework for the theoretical analysis of FV. A perfect illustration of this progress is the design and mathematical analysis of Discrete Duality Finite Volumes (DDFV, for short). We propose now a new class of DDFV for 2nd order elliptic equations involving discontinuous diffusion coefficients or nonlinearities. A one-dimensional linear elliptic equation is addressed here for illustrating the ideas behind our numerical strategy. The algebraic structure of the discrete system we have got is different from that of standard DDFV. The main novelty is that the so-called diamond mesh elements are confined in homogeneous zones for flow problems governed by piecewise constant coefficients. This is got from our judicious definition of the primal mesh. The gain is that there is no need to compute homogenized coefficients to be allocated to the so-called diamond cells as required to conventional DDFV. Notice that poor homogenized permeability allocated to diamond elements leads to poor approximations of fluxes across grid-block interfaces. Moreover for 1-D flow problems in a porous medium involving permeability discontinuities (piecewise constant permeability for instance) the proposed FV scheme leads to a symmetric positive-definite discrete system that meets the discrete maximum principle; we have shown its second order convergence under relevant assumptions.

A Dempster–Shafer Enhanced Framework for Urban Road Planning Using a Model-Based Digital Twin and MCDM Techniques

Rapid urbanization in developing countries presents a critical challenge in the need for extensive and appropriate road expansion, which in turn contributes to traffic congestion and air pollution. Urban areas are economic engines, but their efficiency and livability rely on well-designed road networks. This study proposes a novel approach to urban road planning that leverages the power of several innovative techniques. The cornerstone of this approach is a digital twin model of the urban environment. This digital twin model facilitates the evaluation and comparison of road development proposals. To support informed decision-making, a multi-criteria decision-making (MCDM) framework is used, enabling planners to consider various factors such as traffic flow, environmental impact, and economic considerations. Spatial data and 3D visualizations are also provided to enrich the analysis. Finally, the Dempster–Shafer theory (DST) provides a robust mathematical framework to address uncertainties inherent in the weighting process. The proposed approach was applied to planning for both new road constructions and existing road expansions. By combining these elements, the model offers a sustainable and knowledge-based approach to optimize urban road planning. Results from integrating weights obtained through two weighting methods, the Analytic Hierarchy Process (AHP) and the Bayesian best–worst Method (B-BWM), showed a very high weight for the “worn-out urban texture” criterion and a meager weight for “noise pollution”. Finally, the cost path algorithm was used to evaluate the results from all three methods (AHP, B-BWM, and DST). The high degree of similarity in the results from these methods suggests a stable outcome for the proposed approach. Analysis of the study area revealed the following significant challenge for road planning: 35% of the area was deemed unsuitable, with only a tiny portion (4%) being suitable for road development based on the selected criteria. This highlights the need to explore alternative approaches or significantly adjust the current planning process.

Closed-loop supply chain network of end-of-life wind turbines mathematical model proposal considering environmental, employment, and cost reduction aspects

The worldwide climate issue makes wind energy an attractive renewable energy source. Wind energy reduces carbon emissions, energy prices, creates jobs, improves power access, and ensures energy security. The mass of employed materials in wind turbines indicates a lot of potential raw materials when the whole mass is evaluated. Additionally, a new wind turbine costs $2–$4 million. Remanufacturing or using recycled raw materials from end-of-life wind turbines is practical due to the decreased cost of new wind turbines. To ensure sustainability, CO2 emissions and social advantages like employment must be considered. Given the typical life expectancy of wind turbines at 20–25 years, this issue must be addressed globally in the mid-term. The objective of the conducted study is to provide a multi-objective mathematical framework that encompasses the environmental, social, and economic aspects of wind turbines at the end of their operational lifespan. Hence, this study presents a methodical framework for achieving sustainability in the wind energy sector through the implementation of a closed-loop supply chain. Furthermore, a lexicographic solution approach is offered for each objective of the mathematical model, with the aim of providing decision makers with a comprehensive overview of solutions. The methodology employed in this work involves the utilization of a mixed-integer linear programming (MILP) approach to address the closed-loop supply chain for end-of-life wind turbines. Additionally, a lexicographic solution approach is provided to effectively solve the problem at hand. The novelty of this article lies in its examination of the closed loop supply chain network for end-of-life wind turbines, encompassing the social, environmental, and economic dimensions. This study represents the first of its kind to explore these aspects comprehensively. The proposed mathematical model has three objectives which cover the remanufacturing and recycling and aims to minimize CO2 emissions, unemployment, and transportation cost. In order to evaluate the proposed model, Türkiye is utilized as case with real-life data and Gurobi 9.1.2 is utilized. The locations of wind turbines could not be collected individually, and Euclidian distance was used instead of road distance due to a lack of data. These are the main limitations of the proposed study. The study revealed that the TEC and TCE lexicographic approach has the best recycling and remanufacturing cost-to-manufacturing cost ratio, 52.54 %. The least desired result is 74.50 % from CTE.

Mathematical Framework for Enhancing Machine Failure Prediction in Aviation and Beyond: Leveraging Deep Convolutional Neural Networks and Visual Analytic

Mathematical Framework for Enhancing Machine Failure Prediction in Aviation and Beyond: Leveraging Deep Convolutional Neural Networks and Visual Analytic

Interpreting effective energy barriers to membrane permeation in terms of a heterogeneous energy landscape

Major efforts in recent years have been directed towards understanding molecular transport in polymeric membranes, in particular reverse osmosis and nanofiltration membranes. Transition-state theory is an increasingly common approach to explore mechanisms of transmembrane permeation with molecular details, but most applications of this theory treat all free energy barriers to transport within the membrane as equal. This assumption neglects the inherent structural and chemical heterogeneity in polymeric membranes. In this work, we expand the transition-state theory framework to include distributions of membrane free energy barriers. Our mathematical framework is mechanism-agnostic, such that it generalizes to transport through any membrane for molecular separation. However, we focus our analysis on dense nanofiltration and reverse osmosis membranes. We show that the highest free energy barriers along the most permeable paths, rather than typical paths, provide the largest contributions to the experimentally-observed effective free energy barrier. We show that even moderate, random heterogeneity in molecular barriers will significantly impact how we interpret the mechanisms of transport through these membranes. Our study suggests that experimentally-measured barriers are not easily related to the underlying mechanisms governing transport, and simplified interpretations of these barriers will likely miss the mechanisms most relevant to the overall permeability.

Convection rather than diffusion for fast efficient mRNA vaccine purification

In the rush to produce mRNA vaccines and ensure integrity and purity, capital-intensive purification steps were used, typically involving slow diffusion-driven, inefficient, and costly chromatographic resin processes. We present the first results that microporous affinity synthetic polymer membranes can outperform commercial monolithic and resin approaches for capture of mRNA, thereby promoting fast purification of mRNA vaccines and therapeutics. Our seminal finding is that capture of Fluc-mRNA by synthetic affinity membranes enables efficient intact mRNA recovery (∼100 % of bound mass), faster processing times and lower dispersion by leveraging convective flow with short diffusional path lengths. We also demonstrate higher ligand accessibility and device productivity. Accessible ligand per device volume was 87–94 %, ∼74 % and ∼53 % for the membranes, monoliths, and resin column, respectively. A calibrated mathematical framework predicted mRNA pulse breakthrough with dispersion that was measured in the process tubing and membrane module using a non-reactive tracer and modeled using ideal reactors.