Results in Nonlinear Analysis

Federated multi-modal learning for cross-platform image computation: A functional analysis and nonlinear optimization approach to privacy preservation

2026-01-06T08:12:11+03:00

In Federated multi-modal learning, raw data is not concentrated in a single location because it can perform distributed image computation on heterogeneous platforms. Nonetheless, it is still open to guarantee that the convergence, stability and privacy properties of such systems are mathematically rigorous. In this paper, a functional-analytic, nonlinear-optimization system of federated cross-platform image computation is developed in which local image modalities, and global learning goals are posed as nonlinear variational problems, with local image modalities modelled as an element of separable Hilbert spaces. We present a Nonlinear Federated Proximal Operator (NFPO) that provides a privacy limiting functionality by a dual functional mechanism. We prove existence and uniqueness results of the global minimizer in the presence of coercivity and strong monotonicity, convergence of the NFPO in a contractive mapping argument, and test the framework on synthetic multimodal image datasets given across a plurality of virtual platforms. Numerical experiments show that the proposed approach provides better privacy guarantees with the competitive reconstruction and classification performance. This paper introduces a mathematical based theoretical foundation of a privacy-
conserving federated image computation to cross-platform and multi-modal imaging systems.

Machine learning-enhanced nonlinear differential equation model for predicting osteoporosis progression using bone density imaging data

2026-01-06T07:56:57+03:00

Osteoporosis refers to a chronic bone disease that is characterised by bone loss, microarchitectural loss and high likelihood of getting fragility fracture. Proper forecasting of disease in order to intervene early and plan therapy is crucial. The current research will develop a hybrid modelling system that combines machine learning with nonlinear differential equations to predict the development of osteoporosis through longitudinal bone density imaging. A model of nonlinear bone remodelling is derived based on the coupled system of osteoclast and osteoblast functions, the parameters of the resorption and formation process are adaptively determined with the help of machine learning. External inputs include imaging biomarkers of DXA, QCT and HR-pQCT scans which are used to calibrate
patient-specific remodelling behaviour. It is also extended to a neural differential equation module that is designed to improve the faithfulness of prediction by learning nonlinearities of higher-order that are not modelled by classical physiology-based equations. On of the longitudinal bone imaging dataset, experiments show that the hybrid model has a high prediction accuracy, which decreases
the mean absolute BMD error by 23% relative to standalone ML models and 31 relative to classical ODE models. Noise, missing modalities and variation in the follow-up interval The robustness testing demonstrates that there is negligible predictive power loss with robustness testing. These results imply the possibility of the machine-learning-enhanced nonlinear models yielding predictions on osteoporosis progression that could be used in practise.

Optimized machine learning models for water quality prediction: Integrating support vector machines and random forest through nonlinear functional analysis

2026-01-14T16:35:38+03:00

It is imperative to predict water quality accurately to monitor the environment, human health, and smart water management. Conventional empirical evaluation techniques are inadequate in the description of nonlinear relationships of physicochemical parameters (pH, dissolved oxygen, turbidity, nitrate concentration, and conductivity). This paper hypothesizes a streamlined hybrid machine learning model, which combines Support Vector Machines (SVM) with Random Forest (RF) with nonlinear functional analysis to add predictive accuracy. The model uses nonlinear mapping of kernels, ranking of the importance of variables and functional decomposition to approximate interactions involving complex parameters. It also introduces a multi-stage optimization process that incorporates grid search and cross-validation with nonlinear functional transformation in order to find the best hyperparameters to use in SVM and RF models. Multiyear datasets (collected at freshwater sources) were experimented, and it was found that predictive accuracy improved significantly, with the hybrid model showing that the RMSE was 14.2% lower, and the Pearson correlation coefficient was 9.1 times higher than baseline ML models. The analysis of feature sensitivity and functional interaction demonstrates that nutrient load and dissolved oxygen have a strong nonlinear relationship, which confirms the potential of the proposed framework to represent the ecological relationships. These results indicate that nonlinear functional analysis can allow more consistent and interpretable machine learning models to predict water quality to support the environmental monitoring system in a sustainable way.

Coefficient Estimates and Geometric Analysis of a New Bi-Univalent Function Class

2025-11-26T11:35:54+03:00

In this research, we establish two recently formulated subclasses belonging to the bi-univalent analytic functions characterized by the generalized Sălăgean operator The first subclass consisting of bi-univalent functions within the unit disk and its extended form of subclass are defined by specific argument conditions involving parameters and . Using the concept of subordination and functions with positive real part, the initial coefficients and are estimated with established limits. The results generalize and unify several existing analytic and bi-univalent function’s subclasses. Special cases are discussed to demonstrate the significance and sharpness of the obtained estimates. Additionally, we also analyse the geometric behaviour of functions under this new operator.

Algebraic properties and operator analysis of penta-partitioned intuitionistic neutrosophic soft sets for pattern recognition applications

2026-01-27T12:18:53+03:00

Complex decision-making problems are uncertain, contradictory and partially ignorant conditions that surpass the representational limits of traditional fuzzy, intuitionistic and neutrosophic soft set models. To overcome these limitations, this paper introduces the penta partitioned intuitionistic neutrosophic soft set (PPINSS), a five component extension that distinctly represents truth, indeterminacy, contradiction, falsity and ignorance. Unlike previous models, PPINSS incorporates an intuitionistic dependency between truth and falsity through the balance relation p =1 -t -f , ensuring a coherent and bounded characterization of uncertainty. Fundamental set-theoretic operations such as complement, subset and equality are formally defined and their algebraic properties are rigorously established. Moreover, a comprehensive family of operators—including necessity () and possibility () transformations, aggregation operators (Å,) and parametric mappings m n m n D , F , ( , ) is introduced to model dynamic uncertainty and interdependent reasoning. These operators are shown to satisfy algebraic consistency, duality and closure within the PPINSS domain. By integrating the intuitionistic interdependence of truth and falsity with a fifth component representing latent ignorance, PPINSS offers a unified, logically coherent and semantically rich framework for modeling complex real-world uncertainties. It serves as an effective analytical foundation for multi-criteria decision analysis, distributed intelligence and cognitive reasoning systems.

Mathematical Inequalities for Optimization and Decision-Making in Engineering and Physical Sciences

2025-11-01T23:26:47+03:00

The primary objective of this article is to provide an accessible exposition of the dynamic and evolving field of mathematical inequalities, with a particular emphasis on their role in optimization and decision-making within engineering and the physical sciences. We begin with an elementary introduction to fundamental inequalities, followed by a range of illustrative examples that span from classical applications to contemporary research challenges. To demonstrate the breadth and utility of inequalities, we explore examples from diverse areas of mathematics, including the ubiquitous triangle inequality, which arises in contexts ranging from Euclidean geometry to matrix norms. We highlight key results, such as the interlacing of roots of orthogonal polynomials, which are elegantly formulated through inequality frameworks. In number theory, we present select theorems and conjectures—particularly from the active domain of prime gaps—that are naturally expressed using inequalities. The article also examines inequalities in physics, such as the Clausius inequality in thermodynamics, constraints on electron localization in atomic structures, and bounds related to the cardinality of resistor networks. Overall, this paper aims to introduce readers to the techniques and significance of mathematical inequalities, while showcasing their applications in optimization, theoretical analysis, and practical decision-making across multiple scientific and engineering domains.