Comprehensive Mathematical Framework Research Articles

Mathematical modeling of the synergistic interplay of radiotherapy and immunotherapy in anti-cancer treatments.

While radiotherapy has long been recognized for its ability to directly ablate cancer cells through necrosis or apoptosis, radiotherapy-induced abscopal effect suggests that its impact extends beyond local tumor destruction thanks to immune response. Cellular proliferation and necrosis have been extensively studied using mathematical models that simulate tumor growth, such as Gompertz law, and the radiation effects, such as the linear-quadratic model. However, the effectiveness of radiotherapy-induced immune responses may vary among patients due to individual differences in radiation sensitivity and other factors. We present a novel macroscopic approach designed to quantitatively analyze the intricate dynamics governing the interactions among the immune system, radiotherapy, and tumor progression. Building upon previous research demonstrating the synergistic effects of radiotherapy and immunotherapy in cancer treatment, we provide a comprehensive mathematical framework for understanding the underlying mechanisms driving these interactions. Our method leverages macroscopic observations and mathematical modeling to capture the overarching dynamics of this interplay, offering valuable insights for optimizing cancer treatment strategies. One shows that Gompertz law can describe therapy effects with two effective parameters. This result permits quantitative data analyses, which give useful indications for the disease progression and clinical decisions. Through validation against diverse data sets from the literature, we demonstrate the reliability and versatility of our approach in predicting the time evolution of the disease and assessing the potential efficacy of radiotherapy-immunotherapy combinations. This further supports the promising potential of the abscopal effect, suggesting that in select cases, depending on tumor size, it may confer full efficacy to radiotherapy.

Computer-aided urban energy systems cyber attach detection and mitigation: Intelligence hybrid machine learning technique for security enhancement of smart cities

This paper introduces a sophisticated computer-aided system for enhancing the security of smart cities through urban cyber attack detection and mitigation. The proposed system employs a comprehensive mathematical framework that models the dynamics of cyber attacks, detection, and mitigation, capturing the intricate relationships within a smart city environment. Leveraging a hybrid approach combining Transfer Learning and Meta-Learning, the system adapts to evolving threats and tasks, showcasing superior performance in comparison to standalone methods. Simulation results demonstrate the system’s robustness, with low False Positive and False Negative Rates, high System Security, and effective Mitigation Strategies. This research contributes a state-of-the-art solution for securing smart cities, leveraging intelligent hybrid machine learning techniques.

A multiparameter mathematical model for the problem of nonlinear filtration of fluids in two-layer media

In the realm of fluid dynamics and porous media, the nonlinear filtration of fluids in two-layer media presents a complex and intriguing challenge that demands a comprehensive mathematical framework for accurate representation and analysis. This paper introduces a multiparameter mathematical model designed to elucidate the intricacies of fluid filtration in heterogeneous environments, specifically focusing on two-layer media. The authors research the construction of a multiparameter mathematical model and numerical solution of nonlinear problems of the theory of filtration of unstructured and structured fluids in hydrodynamically connected multilayer formations with different characteristics. The developed models and solution algorithms were tested on hypothetical data.

Redesigning product line for integrated manufacturer-supplier ecosystem in a centralized supply chain: Case of an industrial consumer product

To produce quality products rapidly to derive competitive advantage, the original equipment manufacturer (OEM) must work closely with its suppliers in product redesign processes. Previous literature has emphasized that quality products can be produced effectively and will provide better customer service if product redesign is performed considering the broad OEM-supplier(s) ecosystem. To this end, we seek to develop a mathematical model focused on maximizing the objectives of the OEM-supplier(s) ecosystem considering a centralized supply chain in such a manner that operational-level financial objectives are mapped in terms of product line design decisions and attribute-level assignments. By including operational-level supply chain considerations such as service levels and consumers' preferences, we seek to capture key aspects related to customers’ requirements. This research links product-level technical decisions with enterprise-level business objectives, thus bridging engineering and enterprise-level financial nuances. The problem for the manufacturer pertains to the minimization of the manufacturing cost and maximization of the product premium. The problem for the suppliers pertains to maximization of salvage value, minimization of shortage cost, and maximization of sales revenue. These two problems are then sewed together, and a chemical reaction optimization (CRO) based solution methodology has been deployed. To solve and analyze the devised problem, a case study based on an industrial consumer product is conducted. We evolve a comprehensive mathematical framework that mimics the facets related to product line redesign encapsulating considerations pertaining to functional level design, supply chain dimensions, and manufacturer-supplier profitability. Several novel findings are reported based on the study. For instance, we demonstrate that product configurations for respective segments after the redesign process are contingent upon the service level. Further, we demonstrate that in the industrial consumer product market, higher profit for the OEM-supplier(s) ecosystem is achieved at a slightly lower service level as opposed to a higher theoretical service level. The study has important implications for design and product management.

Some well known inequalities on two dimensional convex mappings by means of Pseudo $ \mathcal{L-R} $ interval order relations via fractional integral operators having non-singular kernel

<abstract><p>Fractional calculus and convex inequalities combine to form a comprehensive mathematical framework for understanding and analyzing a variety of problems. This note develops Hermite-Hadamard, Fejér, and Pachpatte type integral inequalities within pseudo left-right order relations by applying fractional operators with non-singular kernels. Recently, results have been developed using classical Riemann integral operators in addition to various other partial order relations that have some defects that we explained in literature in order to demonstrate the unique characteristics of pseudo order relations. To verify the developed results, we constructed several interesting examples and provided a number of remarks that demonstrate that this type of fractional operator generalizes several previously published results when different things are set up. This work can lead to improvements in mathematical theory, computational methods, and applications across a wide range of disciplines.</p></abstract>

Vibrational Analysis of Constrained Molecular Systems.

Vibrational spectroscopy, including infrared (IR), Raman spectroscopy, and vibrational circular dichroism, is instrumental in determining the structure and composition of molecules. These techniques are highly sensitive to molecular conformations. However, full molecular optimization, necessary for theoretical vibrational spectra, can lead to unintended conformational changes, especially in large biomolecules like polypeptides. To address this, dihedral angle constraints can be imposed during optimization to preserve the molecule's native conformation. Constraint-optimized molecular geometries, not being true stationary points in the full configurational space, pose challenges for traditional vibrational analysis. We address this by considering such geometries as subspace minima, reformulating vibrational analysis to incorporate constraints. Normal modes and spectra consistent with these constraints are obtained by projecting the force constant matrix onto a space orthogonal to the constrained coordinates. This method, illustrated by the example of enkephalin, yields 3N - 6 - m nonzero frequencies after constraint projection, demonstrating its applicability to biomolecules with flexible conformations. Our approach offers a comprehensive mathematical framework to compute vibrational spectra of molecules with conformationally flexible subunits under environmental constraints.

An exploratory framework to urban flood collaborative mitigation strategy considering synergistic effect of inundation volume

As a result of climate change and urbanization, the occurrence of urban flooding has become a common and severe natural hazard that can have devastating consequences on both human society and urban infrastructure. The increasing incidence of urban floods emphasizes the importance of implementing reliable strategies aimed at minimizing losses caused by these disasters. In this regard, an exploratory framework to urban flood collaborative mitigation strategy is developed to optimize the benefits of disaster management and economic investments in flood-prone areas. The proposed framework begins by applying an urban flood model coupled with the source tracking method to simulate hydrological and hydrodynamic process during a storm event, and effectively identifies the synergistic effect of flood volume between different catchments. Subsequently, based on the simulation data from the source tracking model, a method for the comprehensive analysis of spatial prioritization in urban flood management has been developed by employing the combined weight analysis approach. Furthermore, a drainage system dynamic cooperative allocation model is formulated within a comprehensive mathematical programming framework, aiming to effectively reduce the flood volume by considering the synergistic effect. In this comprehensive model, cooperative game theory is utilized to investigate how the flood volume can be effectively mitigated to achieve optimal economic allocation of local municipality. The application of this framework to the Longkungou drainage system of Haikou City demonstrates its potential as a tool for maximizing the economic benefits of disaster mitigation within a drainage system, while minimizing the potential damage caused by urban floods.

Thermal performance of 3D Darcy-forchheimer porous rectangular wavy enclosures containing a water-Fe3O4 ferro-nanofluid under magnetic fields

In the perennial quest for heightened efficiency in heat transfer applications, the inadequacy of water's thermal properties necessitates the exploration of innovative solutions. Nanofluids, particularly those comprising water-Fe3O4 nanoliquid, emerge as promising candidates for augmenting 3D natural convection and entropy generation within permeable wavy-walled rectangular enclosures. The present investigation employs a comprehensive mathematical framework, incorporating the Navier-Stokes equations, magnetic field considerations, and the intricate Darcy-Forchheimer porous media. Utilizing the Galerkin finite element method (GFEM) within the computational domain of COMSOL software, we resolve the system of coupled nonlinear partial differential equations, meticulously derived through non-dimensionalization. Graphical representations meticulously elucidate the nuanced impacts of Rayleigh and Darcy numbers on flow streamlines, temperature distribution, Nu number, and the proportions of global and local thermal entropy generation, frictional entropy generation, and total entropy generation. Our discernments underscore that elevating Rayleigh and Darcy numbers yields a substantial augmentation exceeding 120% in convection flow, particularly prominent for Da = 10−2 vis-à-vis 10−5 and Ra = 106 in comparison to 103. Intriguingly, the optimal positioning of heating surfaces at the bottom right or left surpasses configurations at the bottom middle by a noteworthy margin of approximately 22%. Moreover, the introduction of the Lorentz force, aligned with gravity, manifests a discernible inhibitory effect on flow dynamics, as evidenced by a notable 15% reduction in irreversibility at a Hartmann number of 100.

Abacus Algorithms: A Pure Mathematical Approach to Ancient Calculation Tools

The abacus is one of the oldest calculating tools still in use today. Despite its simplicity, the bead-based interface allows users to conduct complex mathematical operations through a system of sliding beads along wires or rods. While the physical abacus itself provides an intuitive and visual approach to calculation, the underlying operations rely on fundamental mathematical principles. This paper provides a comprehensive mathematical framework that formally describes the algorithms behind abacus calculations. Beginning with basic abacus configuration, we define key components like rods, beads, and bead values required to model abacus states. We then characterize the core abacus algorithms for addition, subtraction, multiplication, and division through set notation, recurrence relations, and state transition diagrams. Our formalized abacus algorithms leverage concepts from number theory, modular arithmetic, combinatorics, and algebra. In addition to offering new mathematical insights into ancient technologies, our work helps bridge connections between the tangible abacus interface and the abstract algorithms powering it. Through examples and proofs, we show how bead manipulations precisely correspond to mathematical transformations. This level of formalization not only helps explain the effectiveness of the abacus, but also illustrates how even rudimentary calculation tools utilize profound mathematical ideas. Our mathematical abacus framework lays the foundation for further analysis as well as modifications and extensions of the classic abacus approach.

The Structure of Density-Potential Mapping. Part I: Standard Density-Functional Theory.

The Hohenberg-Kohn theorem of density-functional theory (DFT) is broadly considered the conceptual basis for a full characterization of an electronic system in its ground state by just the one-body particle density. Part I of this review aims at clarifying the status of the Hohenberg-Kohn theorem within DFT and Part II at different extensions of the theory that include magnetic fields. We collect evidence that the Hohenberg-Kohn theorem does not so much form the basis of DFT, but is rather the consequence of a more comprehensive mathematical framework. Such results are especially useful when it comes to the construction of generalized DFTs.

Coupled-Cluster theory revisited

In a series of two articles, we propose a comprehensive mathematical framework for Coupled-Cluster-type methods. These methods aim at accurately solving the many-body Schrödinger equation. In this first part, we rigorously describe the discretization schemes involved in Coupled-Cluster methods using graph-based concepts. This allows us to discuss different methods in a unified and more transparent manner, including multireference methods. Moreover, we derive the single-reference and the Jeziorski–Monkhorst multireference Coupled-Cluster equations in a unified and rigorous manner.

Coupled-Cluster theory revisited

In a series of two articles, we propose a comprehensive mathematical framework for Coupled-Cluster-type methods. In this second part, we analyze the nonlinear equations of the single-reference Coupled-Cluster method using topological degree theory. We establish existence results and qualitative information about the solutions of these equations that also sheds light of the numerically observed behavior. In particular, we compute the topological index of the zeros of the single-reference Coupled-Cluster mapping. For the truncated Coupled-Cluster method, we derive an energy error bound for approximate eigenstates of the Schrödinger equation.

Optimal control of bioproduction in the presence of population heterogeneity.

Cell-to-cell variability, born of stochastic chemical kinetics, persists even in large isogenic populations. In the study of single-cell dynamics this is typically accounted for. However, on the population level this source of heterogeneity is often sidelined to avoid the inevitable complexity it introduces. The homogeneous models used instead are more tractable but risk disagreeing with their heterogeneous counterparts and may thus lead to severely suboptimal control of bioproduction. In this work, we introduce a comprehensive mathematical framework for solving bioproduction optimal control problems in the presence of heterogeneity. We study population-level models in which such heterogeneity is retained, and propose order-reduction approximation techniques. The reduced-order models take forms typical of homogeneous bioproduction models, making them a useful benchmark by which to study the importance of heterogeneity. Moreover, the derivation from the heterogeneous setting sheds light on parameter selection in ways a direct homogeneous outlook cannot, and reveals the source of approximation error. With view to optimally controlling bioproduction in microbial communities, we ask the question: when does optimising the reduced-order models produce strategies that work well in the presence of population heterogeneity? We show that, in some cases, homogeneous approximations provide remarkably accurate surrogate models. Nevertheless, we also demonstrate that this is not uniformly true: overlooking the heterogeneity can lead to significantly suboptimal control strategies. In these cases, the heterogeneous tools and perspective are crucial to optimise bioproduction.

Fully automated analysis of NMR data with Bayesian nonparametric statistical learning.

The characterization of individual peaks in NMR spectra is an important but challenging step in the investigation of complex biomolecular systems. The major challenge arises due to peak identification which, conventionally, occurs prior to peak characterization. Unluckily, the identification of uncharacterized peaks is highly problematic because of noise and, most importantly, substantial overlap with nearby peaks. For this reason, often the analysis of NMR spectra proceeds by first relying on experts’ opinion to identify peaks and subsequent parameter fitting. Mathematically, peak identification is a problem of model selection and, within a nonparametric analysis framework, can be avoided altogether, this way eliminating the need for experts’ input. In this study, we apply Bayesian nonparametric statistics to develop a comprehensive mathematical framework for automated peak identification and characterization. Our approach relies on advanced statistical representations of FIDs and allows for: simultaneous identification and characterization of individual peaks in an NMR spectrum, accurate noise deconvolution, and extensions to multidimensional NMR or incorporation of specialized protocols. We show that our methods can correctly identify and characterize overlapping peaks without user input. We benchmark our analyses against ground truth peak characteristics using synthetic as well as experimental datasets.

High-power electric vehicle charging: Low-carbon grid integration pathways with stationary lithium-ion battery systems and renewable generation

The energy transition in the mobility sector is well underway. The electrification of road transport is resulting in a shift of the energy demand from the oil and gas sector to the electricity grid. Increasingly aggressive targets for low charging times for Electric Vehicles (EVs) are slated to raise the demand for High-Power Charging (HPC). This is likely to lead to bottlenecks and overloading in vulnerable sections of the electricity grid. Battery Assistance (BA) is a promising grid integration measure for High-Power Charging (HPC) to mitigate these problems. As decarbonization is the primary objective of the energy transition, the determination and comparison of the Global Warming Potential (GWP) footprints for HPC stations with BA is crucial. A comprehensive mathematical framework for the modelling and quantification of GWP footprints for HPC has been developed. The Levelized Emissions of Energy Supply (LEES) methodology has been extended and generalized to handle energy from the grid. A new state variable for the Battery Energy Storage System (BESS) — the State of Carbon Intensity (SOCI) has been introduced to calculate the operation phase GWP footprint of the BESS. The energy consumption GWP footprint for the load is also described by a new quantity — the Load Energy Consumption (LEC) emissions. The effect of incorporation of local Photovoltaic Solar (PV) generation in the energy flows is also investigated. An optimized Energy Management System (EMS) strategy with rolling horizon optimization to minimize emissions has been implemented to regulate energy flows in scenarios with BA and local PV generation. The Levelized Emissions of Energy Supply (LEES) values are obtained for all simulated scenarios and compared against a baseline rule-based EMS strategy. In combination with on-site PV generation, BA could achieve a reduction of 24% in the LEES vis-á-vis the baseline strategy. For reference, two scenarios with Grid Reinforcement (GR) for the grid section with and without local PV generation have also been simulated. With Grid Reinforcement (GR), a reduction of over 2% can be achieved with respect to the baseline EMS strategy for BA. Grid Reinforcement (GR) in conjunction with local PV generation can bring about a further reduction of about 6% with respect to the baseline EMS strategy for BA.

Quantum violation of average causal effects in multiple measurement settings

Estimating causal relations from observed correlations is a central content of science. Although a comprehensive mathematical framework has been developed to identify cause and effect, it is well known that such methods and techniques are not applicable to quantum systems due to Bell's theorem. Generally, the incompatibility between quantum correlation and classical causal theory is illustrated by the violation of Bell inequality. Gachechiladze provides a new method called the violation of lower bound of average causal effect (ACE) to witness the incompatibility. We consider a new lower bound of ACE derived by Cao in multiple measurement settings. We show that there are always pure entangled states and incompatible projective measurements that can generate correlations that violate this new classical lower bound. In Gachechiladze's work, the measurement settings are binary, while the measurement settings in our paper are $K$-ary ($K\ensuremath{\ge}2$) and we give more general conclusions. When $K=2$, the same result as in Gachechiladze's work can be obtained.

Optimal data backup policies for information systems subject to sudden failure

PurposeWith the irresistible growth in digitization, data backup policies become essential more than ever for organizations seeking to improve reliability and availability of organizations' information systems. However, since backup operations do not come free, there is a need for a data-informed policy to decide how often and which type of backups should be taken. In this paper, the authors present a comprehensive mathematical framework to explore the design space for backup policies and to optimize backup type and interval in a given system. In the authors' framework, three separate cost factors related to the backup process are identified: backup cost, recovery cost and data loss cost. The objective function has a multi-criteria structure leading to a backup policy minimizing a weighed function of these factors. To formalize the cost and objective functions, the authors get help from renewal theory in reliability modeling. The authors' optimization framework also formulates mixed policies involving both full and incremental backups. Through numerical examples, the authors show how the authors' optimization framework could facilitate cost-saving backup policies.Design/methodology/approachThe methodology starts with designing different backup policies based on system parameters. Each constructed policy is optimized in terms of backup period using renewal theory. After selecting the best back-up policy, the results are demonstrated through numerical studies.FindingsData backup polices that are tailored to system parameters can result in significant gains for IT (Information Technology) systems. Collecting the necessary parameters to design intelligent backup policies can also help managers understand managers' systems better. Designed policies not only provides the frequency of back up operations, but also the type of backups.Originality/valueThe original contribution of this study is the explicit construction and determination of the best backup policies for IT systems that are prone to failure. By applying renewal theory in reliability, the authors present a mathematical framework for the joint optimization of backup cost factors, i.e. backup cost, recovery time cost and data loss cost.

Integrated computational and in vivo models reveal Key Insights into macrophage behavior during bone healing.

Myeloid-derived monocyte and macrophages are key cells in the bone that contribute to remodeling and injury repair. However, their temporal polarization status and control of bone-resorbing osteoclasts and bone-forming osteoblasts responses is largely unknown. In this study, we focused on two aspects of monocyte/macrophage dynamics and polarization states over time: 1) the injury-triggered pro- and anti-inflammatory monocytes/macrophages temporal profiles, 2) the contributions of pro- versus anti-inflammatory monocytes/macrophages in coordinating healing response. Bone healing is a complex multicellular dynamic process. While traditional in vitro and in vivo experimentation may capture the behavior of select populations with high resolution, they cannot simultaneously track the behavior of multiple populations. To address this, we have used an integrated coupled ordinary differential equations (ODEs)-based framework describing multiple cellular species to in vivo bone injury data in order to identify and test various hypotheses regarding bone cell populations dynamics. Our approach allowed us to infer several biological insights including, but not limited to,: 1) anti-inflammatory macrophages are key for early osteoclast inhibition and pro-inflammatory macrophage suppression, 2) pro-inflammatory macrophages are involved in osteoclast bone resorptive activity, whereas osteoblasts promote osteoclast differentiation, 3) Pro-inflammatory monocytes/macrophages rise during two expansion waves, which can be explained by the anti-inflammatory macrophages-mediated inhibition phase between the two waves. In addition, we further tested the robustness of the mathematical model by comparing simulation results to an independent experimental dataset. Taken together, this novel comprehensive mathematical framework allowed us to identify biological mechanisms that best recapitulate bone injury data and that explain the coupled cellular population dynamics involved in the process. Furthermore, our hypothesis testing methodology could be used in other contexts to decipher mechanisms in complex multicellular processes.

A Cross-Prefetcher Schedule Optimization Methodology

Prefetching offers the potential to significantly improve performance by speculatively loading application data so that it is available before it is needed. By their very nature, prefetching techniques are application behavior dependant. This implies that no universal prefetching solution exists. A combination of prefetching strategies need to be used to target a diverse set of applications. In this work, we develop the first comprehensive mathematical framework that allows a designer to better understand the prefetching opportunities of an application. We first use dynamic analysis to study the memory access behavior of an application and measure a series of metrics to both identify the optimized schedule, and estimate its achievable performance. To validate our model, we implement and evaluate three different prefetching strategies: helper threads, software prefetching and FPGA prefetching. We show that, for each individual scenario, our framework correctly generates the optimized schedule of prefetches and predicts the performance improvement with an accuracy of more than 95%. Using our framework, developers can choose the best prefetching strategy and parameters for their specific workload and use case.

In silico dynamics of COVID-19 phenotypes for optimizing clinical management

Understanding the underlying mechanisms of COVID-19 progression and the impact of various pharmaceutical interventions is crucial for the clinical management of the disease. We developed a comprehensive mathematical framework based on the known mechanisms of the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) infection, incorporating the renin-angiotensin system and ACE2, which the virus exploits for cellular entry, key elements of the innate and adaptive immune responses, the role of inflammatory cytokines, and the coagulation cascade for thrombus formation. The model predicts the evolution of viral load, immune cells, cytokines, thrombosis, and oxygen saturation based on patient baseline condition and the presence of comorbidities. Model predictions were validated with clinical data from healthy people and COVID-19 patients, and the results were used to gain insight into identified risk factors of disease progression including older age; comorbidities such as obesity, diabetes, and hypertension; and dysregulated immune response. We then simulated treatment with various drug classes to identify optimal therapeutic protocols. We found that the outcome of any treatment depends on the sustained response rate of activated CD8+ T cells and sufficient control of the innate immune response. Furthermore, the best treatment-or combination of treatments-depends on the preinfection health status of the patient. Our mathematical framework provides important insight into SARS-CoV-2 pathogenesis and could be used as the basis for personalized, optimal management of COVID-19.