Mathematical Framework Research Articles

Modelling motion energy in psychotherapy: A dynamical systems approach.

In this study we introduce an innovative mathematical and statistical framework for the analysis of motion energy dynamics in psychotherapy sessions. Our method combines motion energy dynamics with coupled linear ordinary differential equations and a measurement error model, contributing new clinical parameters to enhance psychotherapy research. Our approach transforms raw motion energy data into an interpretable account of therapist-patient interactions, providing novel insights into the dynamics of these interactions. A key aspect of our framework is the development of a new measure of synchrony between the motion energies of therapists and patients, which holds significant clinical and theoretical value in psychotherapy. The practical applicability and effectiveness of our modelling and estimation framework are demonstrated through the analysis of real session data. This work advances the quantitative analysis of motion dynamics in psychotherapy, offering important implications for future research and therapeutic practice.

Scaling Effects of the Weissenberg Number in Electrokinetic Oldroyd-B Fluid Flow Within a Microchannel.

This study attempts to extend previous research on electrokinetic turbulence (EKT) in Oldroyd-B fluid by investigating the relationship between the Weissenberg number ( ) and the second-order velocity structure function ( ) under applied electric fields. Inspired by Sasmal's demonstration in Sasmal (2022) of how heterogeneous zeta potentials induce turbulence above a critical , we develop a mathematical framework linking to turbulent phenomena. Our analysis incorporates recent findings on AC (Zhao & Wang, 2017) and DC (Zhao & Wang 2019) EKT, which have defined scaling laws for velocity and scalar structure functions in the forced cascade region. Our finding shows that and , for a length scale , and , where is a velocity fluctuations quantity and denotes the time relaxation parameter. This work establishes a positive correlation between and turbulent flow phenomena through a rigorous analysis of velocity structure functions, thereby offering a mathematical foundation for building the design and optimization of EKT-based microfluidicdevices.

Constraint Realization‐Based Hamel Field Integrator for Geometrically Exact Planar Euler–Bernoulli Beam Dynamics

ABSTRACTIn this article, we first introduce a Hamel field integrator designed for a geometrically exact Euler–Bernoulli beam with infinite‐dimensional holonomic constraints, constructed using a Lagrange multiplier. This method addresses the complexities introduced by constraints, but the additional multiplier introduces a new degree of freedom and hence results in a system with mixed‐type partial differential equations. To address this issue, we further propose a constraint realization method based on perturbation theory for infinite‐dimensional mechanical systems within the framework of Hamel's formalism. This method circumvents the use of additional Lagrange multiplier, significantly reducing the computational complexity of modeling problems. Building on this, we construct a perturbed Hamel field integrator optimized for parallel computing and incorporate artificial viscosity to accelerate constraint convergence. While applicable to three dimensions, our method is demonstrated in a simplified context using planar Euler–Bernoulli beam examples to illustrate the effectiveness of the unified mathematical framework.

Treatment of inelastic material models within a dynamic ALE formulation for structures subjected to moving loads

AbstractThis article showcases the development of a dynamic Arbitrary Lagrangian Eulerian (ALE) formulation to account for inelastic material models within a finite element framework. Such a formulation is commonly utilized in research domains like fluid mechanics, fluid‐structure interaction, quasi static remeshing techniques, and quasi static load movement. The work at hand describes the application of the ALE formulation to efficiently analyse structures subjected to moving loads in the field of transient inelastic solid mechanics. In particular, structures such as pavements, gantry crane girders etc., which are subjected to moving loads, can be numerically simulated, and their transient response in the relevant region around the load can be obtained without relying on moving loads. The focus of this article is to facilitate the treatment of history variables stemming from inelastic material models. Of particular interest is the advection procedure required to transport the history variables through the mesh, as the material appears to flow through it. The mathematical framework necessary to treat this advection process is described in detail, considering a nonlinear viscoelastic material model on a neo‐Hookean base at finite deformations. Then, four methods for numerically achieving the advection are implemented within a transient finite element ALE formulation. These methods are compared against each other, and additionally with the conventional Lagrangian method for validation. The results demonstrate satisfactory agreement with conventional simulation methods, while offering a significant improvement in terms of computation speed. With the work at hand, the dynamic response of inelastic materials subjected to moving loads can be numerically simulated in a computationally efficient manner.

Quantum networks theory

The formalism of quantum theory over discrete systems is extended in two significant ways. First, quantum evolutions are generalized to act over entire network configurations, so that nodes may find themselves in a quantum superposition of being connected or not, and be allowed to merge, split and reconnect coherently in a superposition. Second, tensors and traceouts are generalized, so that systems can be partitioned according to almost arbitrary logical predicates in a robust manner. The hereby presented mathematical framework is anchored on solid grounds through numerous lemmas. Indeed, one might have feared that the familiar interrelations between the notions of unitarity, complete positivity, trace-preservation, non-signalling causality, locality and localizability that are standard in quantum theory be jeopardized as the neighbourhood and partitioning between systems become both quantum, dynamical, and logical. Such interrelations in fact carry through, albeit two new notions become instrumental: consistency and comprehension.

On the structure of species-function participation in multilayer ecological networks

Understanding how biotic interactions shape ecosystems and impact their functioning, resilience and biodiversity has been a sustained research priority in ecology. Yet, traditional assessments of ecological complexity typically focus on species-species interactions that mediate a particular function (e.g., pollination), overlooking both the synergistic effect that multiple functions might develop as well as the resulting species-function participation patterns that emerge in ecosystems that harbor multiple ecological functions. Here we propose a mathematical framework that integrates various types of biotic interactions observed between different species. Its application to recently collected data of an islet ecosystem—reporting 1537 interactions between 691 plants, animals and fungi across six different functions (pollination, herbivory, seed dispersal, decomposition, nutrient uptake, and fungal pathogenicity)—unveils a non-random, nested structure in the way plant species participate across different functions. The framework further allows us to identify a ranking of species and functions, where woody shrubs and fungal decomposition emerge as keystone actors whose removal have a larger-than-random effect on secondary extinctions. The dual insight—from species and functional perspectives—offered by the framework opens the door to a richer quantification of ecosystem complexity and to better calibrate the influence of multifunctionality on ecosystem functioning and biodiversity.

Heterogeneous survival upon disinfection underlies evolution of increased tolerance.

Disinfection is key to control the spread of infections. But the application of disinfectants bears the risk to promote the evolution of reduced susceptibility to antimicrobials if bacteria survive the treatment. The ability of individual bacteria to survive disinfection can display considerable heterogeneity within isogenic populations and may be facilitated by tolerant persister subpopulations. Using time-kill kinetics and interpreting the data within a mathematical framework, we quantify heterogeneity and persistence in Escherichia coli when exposed to six different disinfectants. We find that the level of persistence, and with this the risk for disinfection failure, depends on the disinfectant. Importantly, evolution experiments under recurrent disinfection provide evidence that links the presence of persisters to the ability to evolve reduced susceptibility to disinfectants. This study emphasizes the impact of heterogeneity within bacterial populations on disinfection outcomes and the potential consequences for the evolution of antimicrobial resistances.

Tangent space functional reconfigurations in individuals at risk for alcohol use disorder

Abstract Human brain function dynamically adjusts to ever-changing stimuli from the external environment. Studies characterizing brain functional reconfiguration are nevertheless scarce. Here we present a principled mathematical framework to quantify brain functional reconfiguration when engaging and disengaging from a stop signal task (SST). We apply tangent space projection (a Riemannian geometry mapping technique) to transform functional connectomes (FCs) of 54 participants and quantify functional reconfiguration using the correlation distance of the resulting tangent-FCs. Our goal was to compare functional reconfigurations in individuals at risk for alcohol use disorder (AUD). We hypothesized that functional reconfigurations when transitioning to/from a task would be influenced by family history of alcohol use disorder (FHA) and other AUD risk factors. Multilinear regression models showed that engaging and disengaging functional reconfiguration were associated with FHA and recent drinking. When engaging in the SST after a rest condition, functional reconfiguration was negatively associated with recent drinking, while functional reconfiguration when disengaging from the SST was negatively associated with FHA. In both models, several other factors contributed to the functional reconfiguration. This study demonstrates that tangent-FCs can characterize task-induced functional reconfiguration, and that it is related to AUD risk.

Resilient Coastal Protection Infrastructures: Probabilistic Sensitivity Analysis of Wave Overtopping Using Gaussian Process Surrogate Models

This paper presents a novel mathematical framework for assessing and predicting the resilience of critical coastal infrastructures against wave overtopping hazards and extreme climatic events. A probabilistic sensitivity analysis model is developed to evaluate the relative influence of hydrodynamic, geomorphological, and structural factors contributing to wave overtopping dynamics. Additionally, a stochastic Gaussian process (GP) model is introduced to predict the mean overtopping discharge from coastal defences. Both the sensitivity analysis and the predictive models are validated using a large homogeneous dataset comprising 163 laboratory and field-scale tests. Statistical evaluations demonstrate the superior performance of the GPs in identifying key parameters driving wave overtopping and predicting mean discharge rates, outperforming existing regression-based formulae. The proposed model offers a robust predictive tool for assessing the performance of critical coastal protection infrastructures under various climate scenarios.

Balancing the scales: assessing the impact of irrigation and pathogen burden on potato blackleg disease and soil microbial communities

BackgroundUnderstanding the interaction between environmental conditions, crop yields, and soil health is crucial for sustainable agriculture in a changing climate. Management practices to limit disease are a balancing act. For example, in potato production, dry conditions favour common scab (Streptomyces spp.) and wet conditions favour blackleg disease (Pectobacterium spp.). The exact mechanisms involved and how these link to changes in the soil microbiome are unclear. Our objectives were to test how irrigation management and bacterial pathogen load in potato seed stocks impact: (i) crop yields; (ii) disease development (blackleg or common scab); and (iii) soil microbial community dynamics.MethodsWe used stocks of seed potatoes with varying natural levels of Pectobacterium (Jelly [high load], Jelly [low load] and Estima [Zero – no Pectobacterium]). Stocks were grown under four irrigation regimes that differed in the timing and level of watering. The soil microbial communities were profiled using amplicon sequencing at 50% plant emergence and at harvest. Generalised linear latent variable models and an annotation-free mathematical framework approach (ensemble quotient analysis) were then used to show the interacting microbes with irrigation regime and Pectobacterium pathogen levels.ResultsIrrigation increased blackleg symptoms in the plots planted with stocks with low and high levels of Pectobacterium (22–34%) but not in the zero stock (2–6%). However, withholding irrigation increased common scab symptoms (2–5%) and reduced crop yields. Irrigation did not impact the composition of the soil microbiome, but planting stock with a high Pectobacterium burden resulted in an increased abundance of Planctomycetota, Anaerolinea and Acidobacteria species within the microbiome. Ensemble quotient analysis highlighted the Anaerolinea taxa were highly associated with high levels of Pectobacterium in the seed stock and blackleg symptoms in the field.ConclusionsWe conclude that planting seed stocks with a high Pectobacterium burden alters the abundance of specific microbial species within the soil microbiome and suggest that managing pathogen load in seed stocks could substantially affect soil communities, affecting crop health and productivity.BWvw4eepooU5poSUC2MUy8Video

PSO-based Quaternion Fourier Transform steganography: Enhancing imperceptibility and robustness through multi-dimensional frequency embedding

This paper presents a novel steganography technique using the Quaternion Fourier Transform (QFT) in the 4D frequency domain to enhance imperceptibility and robustness in digital image embedding. Steganography, the art of hiding information within media, faces challenges in balancing security, imperceptibility, and robustness. To address this, we leverage the multi-dimensional properties of quaternions, enabling the embedding of secret data in both grayscale and color images. For grayscale images, two quaternion dimensions are utilized for intensity and secret data, while for color images, all four dimensions are employed with one reserved for metadata. The research question centers on how to maximize spatial dispersion and color similarity while maintaining high imperceptibility and robustness against attacks. Experimental results show that the proposed method improves visual imperceptibility by more than 4 % and exhibits a 13 % increase in robustness against common steganalysis attacks compared to the best state-of-the-art existing technique. These advancements highlight the potential of this method for applications in secure communication, digital watermarking, and copyright protection. By combining the quaternion mathematical framework with a novel optimization strategy, this approach significantly improves upon traditional steganography methods.

Non-radiating sources in elastodynamics and their applications in the exterior cloaking

In this work, we develop a mathematical framework on constructed general non-radiating sources of elastic waves governed by the Navier equation via the approach of Helmholtz decomposition and potential theory in elastodynamics. Our study offers a rather comprehensive analysis. We first provide a rigorous justification of the general non-radiating sources. Based on the complete destructive interference of external elastic fields generated by specific radiating sources, a general non-radiating elastic source is derived and shown to possess a hidden interior wave field. For an incident wave, targets remain invisible within non-radiating source regions, and the geometry and boundary conditions of obstacles can be very general, which holds significant practical implications. Moreover, we introduce an effective novel method for designing such generalized non-radiating sources. To avoid the complex structure, we propose to use radiating source overlay construction on specific nodes at the boundary of non-radiating regions construction and derive sharp error estimates to evaluate the cloaking performance. The proposed scheme is capable of nearly cloaking arbitrary obstacles with a high accuracy. Numerical verifications validate the precision of our analytical findings.

Study on an Intelligent English Translation Method Using an Improved Convolutional Neural Network Model

This study uses an enhanced convolutional neural network (CNN) model for English translation. Traditional translation methods often struggle with complex language structures, prompting the adoption of deep learning techniques, particularly CNNs, in natural language processing. The research outlines CNN fundamentals and their relevance in language processing, elucidating the design and implementation of an improved CNN model. To address CNN limitations in maintaining coherence during the translation of lengthy texts, historical attention mechanisms are incorporated to enhance translation performance. Experimental validation conducted in MATLAB demonstrates notable improvements in translation task performance, evidenced by significant increases in BLEU scores. Results highlight the model's capacity to integrate contextual information, thereby enhancing translation coherence and accuracy. Additionally, the study establishes a mathematical framework for augmenting CNNs with attention mechanisms, providing valuable insights for the development of intelligent English translation systems.

Inflation-induced motility for long-distance vertical migration.

The vertical migrations of pelagic organisms play a crucial role in shaping marine ecosystems and influencing global biogeochemical cycles. They also form the foundation of what might be the largest daily biomass movement on Earth. Surprisingly, among this diverse group of organisms, some single-cell protists can transit depths exceeding 50m without employing flagella or cilia. How these non-motile cells perform large migrations remains unknown. It has been previously proposed that this capability might rely on the cell's ability to regulate its internal density relative to seawater. Here, using the dinoflagellate algae Pyrocystis noctiluca as a model system, we discover a rapid cell inflation event post cell division, during which a single plankton cell expands its volume 6-fold in less than 10min. We demonstrate this rapid cellular inflation is the primary mechanism of density control. This self-regulated cellular inflation selectively imports fluid less dense than surrounding seawater and can thus effectively sling-shot a cell and reverse sedimentation within minutes. To accommodate its dramatic cellular expansion, Pyrocystis noctiluca possesses a unique reticulated cytoplasmic architecture that enables a rapid increase in overall cell volume without diluting its cytoplasmic content. We further present a generalized mathematical framework that unifies cell-cycle-driven density regulation, stratified ecology, and associated cell behavior in the open ocean. Our study unveils an ingenious strategy employed by a non-motile plankton to evade the gravitational sedimentation trap, highlighting how precise control of cell size and cell density can enable long-distance migration in the open ocean.

A Study of q-Deformed Bosons, and Their Implications to Quantum Optics

In this study, we investigate three types of q-deformed boson oscillators, focusing on their mathematical frameworks and thermodynamic properties. We calculate key thermodynamic quantities, such as internal energy and entropy, as functions of the deformation parameter q. Our results reveal that these oscillators are eigenstates of specific deformed boson annihilation operators. We also analyze their unique characteristics and implications in deformed quantum optics. Furthermore, we examine the impact of q-deformation on qutrit logic gates, including cycle, self-shift, controlled cycle, controlled self-shift, Feynman, ternary Toffoli, and Fredkin gates, highlighting their altered computational properties. This research contributes to a deeper understanding of q-deformed systems and their applications in quantum computing. Overall, it opens new avenues for exploring the interplay between deformation parameters and quantum information processing.

Comparison of four light-response models using relative curvature measures of nonlinearity

Photosynthetic light response curves serve as powerful mathematical tools for quantitatively describing the rate of photosynthesis of plants in response to changes in irradiance. However, in practical applications, the daunting task of selecting an appropriate nonlinear model to accurately fit these curves persists as a significant challenge. Thus, there arises a need for a method to systematically evaluate the efficacy of such models. In the present study, four distinct nonlinear models, namely Exponential Model (EM), Rectangular Hyperbola Model (RHM), Nonrectangular Hyperbola Model (NHM), and Modified Rectangular Hyperbola Model (MRHM), were used to fit the relationship between light intensity and the rate of photosynthesis across 42 empirical datasets. The goodness of fit for each model was assessed using the root-mean-square error, and relative curvature measures of nonlinearity were employed to assess the nonlinear behavior of the models. In terms of goodness of fit, pairwise difference tests of the root-mean-square error revealed that there was little to choose among the four models, although RHM gave a marginally poorer fit. However, in terms of nonlinear behavior, EM not only provided the most favorable linear approximation performance at the global level, but also exhibited the best close-to-linear behavior at the individual parameter level among the four models across the 42 datasets. Consequently, the results strongly advocate for EM as the most suitable mathematical framework for fitting photosynthetic light response curves. These findings provide insights into the model assessment for nonlinear regression in describing the relationship between the photosynthetic rate and light intensity.

Fragility estimation for performance-based structural design of floating offshore wind turbine components

This study proposes a computational and mathematical framework aimed at assessing the reliability of structural components within Floating Offshore Wind Turbines (FOWT) that reflects the various sources of uncertainties coupled between structural analyses, hydrodynamics, and aerodynamics. The limit state functions are represented through structural capacity and environmental demand models for selected structural failure modes that incorporate fully coupled aero-hydro-servo-elastic analysis. The fragility surfaces are developed for a selected benchmark wind turbine for both operating and parking conditions. The fragilities are also estimated under 50-year and 100-year environmental conditions in the selected U.S. coastal regions. It is found that the wind speed variations largely affect the fragility during non-operation, while wave height variations are significant during operation. Increased uncertainties in environmental parameters raised failure probabilities, especially in lower fragility ranges targeted by design codes. Analyses in U.S. coastal environments show both parking and operating conditions can be critical, challenging the previous focus on parking. Sensitivity studies reveal that under mild conditions, structural reliability is influenced by moment of inertia and material strength, but as environmental loads increase, these parameters become equally significant. Increased uncertainties in parameters lead to higher failure risks, especially below 25 m/s wind speeds.

Operando Spectroelectrochemical Imaging of Concentration Fields and Tafel Kinetics in Microfluidic Electrochemical Devices.

With the development of microtechnologies for energy conversion and storage, mass transfer and micromolar concentration variations need to be measured at the microscale. These advances need to be accompanied by novel imaging techniques with the capability of achieving high spatial resolution while detecting very small signal variations (less than 0.1%). Thus, in this study, a new microscopy technique is proposed based on a combination of electrochemical impedance spectroscopy (EIS) and visible imaging spectroscopy to measure the concentration fields at the micromolar scale in operando microfluidic fuel cells (MFCs). This technique exploits EIS modulation and Fourier analysis to reduce the noise during concentration field imaging. A mass transfer model in the periodic regime is derived to validate the measurements and to estimate the Tafel kinetics and mass diffusivities during potassium permanganate reduction from only one potential measurement. The proposed imaging method and mathematical framework presented in this study can be used to study binary electrochemical reactions without gas production.

Enhancing Security in Geographic Information Systems: Anonymization and Differential Privacy Techniques for Protecting Sensitive Geospatial Data

As Geographic Information Systems (GIS) increasingly facilitate the analysis and sharing of geospatial data, the protection of sensitive information becomes paramount. This research explores the implementation of anonymization and differential privacy techniques to enhance security in GIS. Anonymization methods effectively remove or obscure personally identifiable information from geospatial datasets, while differential privacy introduces a mathematical framework that allows for the sharing of aggregate data without compromising individual privacy. This study evaluates the strengths and weaknesses of these techniques, demonstrating their effectiveness in maintaining the utility of geospatial data while safeguarding sensitive information. Through case studies and comparative analysis, we provide insights into best practices for integrating these privacy-preserving strategies into GIS applications, ensuring compliance with legal regulations and fostering public trust in geospatial technologies.

Relationship between COVID-19 Cases and Environmental Contaminants in Quito, Ecuador.

The relationship between COVID-19 infections and environmental contaminants provides insight into how environmental factors can influence the spread of infectious diseases. By integrating epidemiological and environmental variables into a mathematical framework, the interaction between virus spread and the environment can be determined. The aim of this study was to evaluate the impact of atmospheric contaminants on the increase in COVID-19 infections in the city of Quito through the application of statistical tests. The data on infections and deaths allowed to identify the periods of greatest contagion and their relationship with the contaminants O3, SO2, CO, PM2.5, and PM10. A validated database was used, and statistical analysis was applied through five models based on simple linear regression. The models showed a significant relationship between SO2 and the increase in infections. In addition, a moderate correlation was shown with PM2.5, O3, and CO, and a low relationship was shown for PM10. These findings highlight the importance of having policies that guarantee air quality as a key factor in maintaining people's health and preventing the proliferation of viral and infectious diseases.