Search Results (4,010)

Authenticated key exchange has been built for decades on a small number of hard mathematical problems, and the elliptic curve discrete logarithm is one of them. The structural complexity of biological macromolecules, however, has rarely been used as a source of secret material in such protocols. In this paper, a protein-derived authenticated key exchange protocol, called PDKE-2, is proposed to address this gap. The protocol keeps elliptic curve Diffie–Hellman at its core and adds two ingredients from structural biology. Firstly, a per-session base point G* is generated by applying RFC 9380 hash-to-curve to a session mapping table. Secondly, a long-term secret shared between the two parties is derived from the SHA-256 digest of an AlphaFold2 inter-residue distance matrix. In contrast to an earlier version of this design, the two parties run ephemeral Diffie–Hellman in each session and the session key is never sent over the channel; only transcript-based confirmation tags are exchanged. Thus, the protocol provides forward secrecy and avoids a confidentiality weakness that existed in the earlier version. The security of PDKE-2 is analysed in the Real-or-Random model, where session key secrecy, forward secrecy and mutual authentication are proved under the Gap Diffie–Hellman, PRF-HKDF and EUF-CMA-HMAC assumptions. In addition, the protocol is modelled in ProVerif against a Dolev–Yao attacker, and the secrecy of the session key, the secrecy of the shared secret and the injective agreement in both directions are all confirmed. A Python reference implementation is also provided. In the implementation, key agreement and mutual authentication succeed in every session, and the complete handshake takes about 15 ms in the reference code. It should be noted that the protein layer does not increase the elliptic curve security level; it works as an authentication secret whose strength depends on the difficulty of guessing the protein sequence when this sequence is kept private. Full article

Fluids whose viscosity and surface tension depend on deformation history are poorly described by integer-order models, and fractional differential equations have been adopted for them across rheology and applied mathematics. Work on this subject remains scattered, and no synthesis has established which operators are in use, how the equations are solved, or what remains undone. This review supplies that synthesis and states the open problems that follow. Under the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) 2020 protocol, 682 records from Scopus, ScienceDirect, and SpringerLink were reduced to 23 eligible articles by deduplication of 61 records, title and abstract screening of 533, and full-text assessment of the 81 retrievable reports. A supplementary search on the term “fractional viscoelastic”, run as a sensitivity analysis, added five more, giving a final corpus of 28 studies from 1986 to 2026, classified by operator, model, solution method, fluid, property, and validation status. The springpot-based Fractional Maxwell Model and the Caputo-derivative dominate, no study applies the Atangana–Baleanu–Caputo or Caputo–Fabrizio operator constitutively, and joint modeling of the two properties is confined to three studies of one lubricating oil. Five open problems are stated, each with its supporting evidence and the direction that would resolve it. Full article

Time-of-flight magnetic resonance angiography (TOF-MRA) is widely used for noninvasive visualization of arterial anatomy, but extracting hemodynamics like blood velocity typically requires supplementary phase-contrast scans, tagging or multi-TE images. This study proposes a novel, physics-informed computational framework to extract variable fluid velocity directly from standard TOF-MRA signal profiles. We analytically expand the approach-to-steady-state Bloch equations to include convective flow, establishing a mathematical relationship between the spatial decay of longitudinal magnetization and fluid velocity. The velocity derivation was further extended to pointwise estimation over a 1-D centerline, overcoming the limitations of constant-velocity assumptions. To validate and solve this problem, a MATLAB (R2025b) simulation framework was developed to model fluid flow in two variable-geometry flowing tube cases, i.e., continuous narrowing and focal stenosis, under synthetic scanner noise. A global inverse optimization approach utilizing Dual-Tikhonov regularization was applied to stably invert the ill-posed transit time integral, actively penalizing high-frequency numerical ringing while preserving structural curves. The computational simulations successfully recovered ground-truth point-wise velocities, tracking gradual hemodynamic accelerations and sharp stenotic jets. This theoretical framework and the example centerline TOF-MRA signal intensity provide a robust mathematical proof-of-concept that quantitative, localized functional hemodynamic metrics can be extracted from standard structural MRA imaging, establishing a foundation for advanced flow quantification without requiring additional scan time. Full article

The derivatives of the inverse of a one-to-one function are needed in a range of applied contexts, from random variate generation and molecular simulation to nuclear physics and bias reduction for maximum likelihood estimates, yet existing treatments derive them by ad hoc, application-specific means without a unifying framework. Here, we give the general derivative of the inverse of a one-to-one function, firstly by a recurrence formula, secondly by repeated differentiation, and thirdly—and most explicitly—in closed form using the partial exponential Bell polynomials associated with Faà di Bruno’s chain rule, providing a single representation that subsumes and extends earlier case-specific results and that can be taken to arbitrary order. We illustrate the practical value of these results in mathematical statistics, applying them to bias reduction for maximum likelihood estimates in one-parameter exponential families, including the gamma shape parameter and canonical regression models. Python programs implementing the recurrence and the Bell polynomial representations are included. Full article

In this paper, the new fractal modified Zakharov–Kuznetsov equation (fmZKe) defined on Cantor sets is investigated. The fmZKe is a non-differentiable model that arises naturally in mathematical physics, nonlinear wave theory, and plasma physics. The extended rational sine–cosine method is utilized to construct new optical soliton solutions of the model. The fmZKe is reduced to a non-differentiable ordinary differential equation by applying a non-differentiable wave transformation defined on Cantor sets. This reduction leads to a system of linear algebraic equations, which upon solving yields several exact solutions of the model. Furthermore, to establish a clear understanding of the model’s behavior, non-smooth graphical representations of the solutions are presented for various parameter values. The stability analysis of the newly obtained solutions in a classical sense is examined using stability theory, and the real-life applications of the results are highlighted. Full article

In this paper, a controlled Picard semi-analytical technique is improved for a branch of nonlinear fractional stochastic delay differential equations since the nonlinear terms always prevent the establishment of closed-form solutions. The proposed approach extends the known Picard iteration by embedding a convergence-control parameter that improves the flexibility and stability of the iterative scheme while keeping the original mathematical formulation. A careful theoretical analysis is presented to establish the existence of the iterative sequence, its mean-square boundedness, convergence, and an explicit error estimate under standard Lipschitz continuity and linear growth assumptions. Moreover, a Numerical Picard implementation is updated to rebuild stochastic sample trajectories and to give an independent illustration through comparison with a predictor–corrector scheme. The presented methodology is applied to fractional stochastic models of human postural sway and logistic population models. The numerical results illustrate that the semi-analytical framework evaluates the expectation of and variance in the stochastic response for various fractional orders accurately. Although the semi-analytical controlled Picard method is incapable of performing a lot of iterations, it generates statistical moments that agree with those obtained using both the Numerical Picard and predictor–corrector methods, while explicit semi-analytical representations of the solution. These results show that the presented technique gives an accurate and effective approach for examining nonlinear fractional stochastic delay systems from biological, ecological, and engineering applications. Full article

As the retail banking sector shifts toward automated lending, the black-box nature of high-performing machine learning models remains a significant barrier to regulatory transparency and institutional trust. A critical gap in existing literature is the lack of deployed frameworks that simultaneously optimize predictive accuracy, manage asymmetric financial risks, and provide actionable interpretability. To bridge this gap, this study aims to develop and evaluate a highly interpretable, ethically accountable ensemble machine learning framework for credit risk assessment. Utilizing a cross-sectional public dataset of over 45,000 generalized retail banking records, this research conducts a comprehensive comparative analysis of four diverse ensemble architectures: Bagging, Boosting, Stacking, and Voting. To address inherent class imbalance and evaluate risk tolerance, the models were integrated with Synthetic Minority Over-sampling Technique (SMOTE) and Adaptive Synthetic Sampling (ADASYN) resampling techniques. While all architectures demonstrated high discriminative power, the SMOTE-balanced Bagging model emerged as the superior performer, achieving a peak Area Under the Curve (AUC) of 0.972 by establishing a safe operational threshold that strictly minimizes costly false approvals. Crucially, a SHapley Additive exPlanations (SHAP) framework was applied across all four models to decode their internal logic. The SHAP analysis successfully validated that the ensembles prioritize core financial behavior, such as default history and loan-to-income ratios, while correctly assigning near-zero predictive weight to demographic traits like gender and education. By empirically proving that high-performance algorithms can be mathematically blind to demographic biases, this framework directly advances SDG 10 (Reduced Inequalities). Furthermore, by resolving the performance-transparency trade-off, this study provides the accountable, feature-level justifications required for secure and sustainable financial inclusion (SDG 8). Full article

Transportation networks are subject to multiple sources of uncertainty, ranging from stochastic fluctuations in demand and travel times to epistemic indeterminacy in infrastructure condition, disruption risk, and user behavior. A diverse body of mathematical frameworks has emerged in response, including stochastic programming and probabilistic reliability analysis, fuzzy and possibilistic approaches, Liu’s uncertainty theory and uncertain programming, and robust or distributionally robust optimization. This article delivers a comprehensive, mathematically oriented synthesis of these paradigms for transportation networks, with emphasis on network-level structures—paths, flows, spanning trees, and network design problems—and on reliability notions including connectivity, travel-time, capacity, and max-type reliability. A central theme is that modelling choices about uncertainty representation and reliability indices are inseparable from questions of stability and sensitivity: how robust are optimal or near-optimal configurations when parameters vary within plausible ranges? Building on deterministic post-optimal analysis, this paper reviews tolerance-based stability concepts for uncertain most reliable paths, maximum reliable transmission paths, and uncertain minimum spanning trees under Liu-type uncertainty and demonstrates how inverse-distribution mappings yield exact deterministic equivalents and belief-based robustness margins. A dedicated comparative framework is developed, summarizing the data requirements, core advantages, typical limitations, and suitable engineering scenarios of each uncertainty paradigm to guide model selection in practice. The discussion extends to practical applications in post-disaster planning, infrastructure investment prioritization, and supply chain network design and identifies open research directions including network-wide travel-time reliability under belief-based uncertainty, unified stability frameworks across paradigms, and the integration of machine learning for uncertainty distribution elicitation. The emphasis throughout is on conceptual structure, modelling assumptions, and interpretability of reliability and stability indices, thereby positioning uncertain transportation networks as a rich interface between applied mathematics, operations research, and infrastructure planning. Full article

Generative Artificial Intelligence (AI) has become a rapidly expanding area of scientific research, generating a growing body of literature across technical and applied domains. This study provides a comprehensive bibliometric analysis of global generative AI research to characterize its publication growth, disciplinary and geographical distribution, institutional participation, funding patterns, citation performance, and thematic development. The analysis covers 22,758 Scopus-indexed journal articles and conference papers published between 2020 and 2025, identified using the phrase “generative artificial intelligence” enclosed in double quotation marks in TITLE-ABS-KEY fields. A reproducible computational workflow was used to examine publication output, document types, subject areas, countries, institutions, funding sponsors, citation patterns, and keyword co-occurrence. Citation analysis incorporated annualized citation rates and cohort-normalized annual citation rates to improve comparisons across publication years. Results show a pronounced acceleration in publication output after 2022, with an approximate 105% compound annual growth rate over 2020–2025. Computer Science remained the largest subject area, while substantial representation extended across Engineering, Social Sciences, Medicine, Mathematics, and other domains. Publication activity was concentrated among leading countries and institutions, with the United States and China recording the highest output. Funding analysis identified major national and international sponsors through publication–sponsor associations. Citation performance varied substantially across cohorts, with the 2023 cohort exhibiting the highest cohort-normalized annual citation rate (1.58). Keyword analysis revealed a thematic shift from early AI and GAN-related research toward generative AI and large language model themes, alongside education, innovation, human–AI interaction, and responsible AI. The findings provide an evidence-based, multidimensional characterization of the rapidly evolving generative AI research landscape. Full article

The paper examines the impact of gas turbine engine component manufacturing quality on the efficiency criteria of its life test. Known methods for selecting test parameters apply maximum damageability equivalence and minimum test time as test efficiency criteria. This study also proposes taking into account the maximization of engine life cycle profits through the proper selection of test parameters. Engine components that determine its life were selected: the turbine blade, rotor bearing, reducer driving gear, fan bearing, and DC and AC generators. Both the mathematical expectation and variance of the quality parameters were varied during the study. The manufacturing quality of engine components and assemblies is characterized by geometric, mechanical, and physical parameters. These parameters include bearing fit diameters, initial radial clearance, turbine blade geometry, mechanical properties and gear shape, generator insulation quality, and others. Selection of parameters was based on the life cycle simulation model. The following results were obtained in the course of the study within the framework of modeling: (1) Manufacturing accuracy has a more significant impact on test results than deviations from mean values of initial state parameters; (2) under the accepted assumptions, despite the fact that variation in production parameters from the standard values does not affect the comparability of test results, they lead to an acceleration of the testing process. At the same time, this entails a decrease in overall economic efficiency throughout the entire life cycle of the product; (3) according to the obtained results, the overall profitability of a production run of engines is primarily determined by the quality characteristics of the turbine blades, and least of all by the fan bearing quality parameters. Full article

Symmetric quantum calculus offers a dynamic framework for investigating the various classes of functions. However, the symmetric quantum operators become inconclusive at certain points. To overcome the limitations of existing calculi, operators over finite intervals have been extensively explored. To develop a more general and applicable setup, we introduce the symmetric quantum derivative and integral operators governed by an arbitrary point. Furthermore, we discuss structural properties of the newly developed operators and special cases to relate to the existing literature. Then, by applying the concepts of general symmetric quantum operators, convexity, Lipschitzian property, and Korkine’s identity, we derive a new set of inequalities, including Hermite–Hadamard, Ostrowski, Hólder, Minkowski, and Gruss-type inequalities, respectively. The proposed inequalities are useful to derive the bounds of generalized symmetric quantum integrals. Furthermore, the newly developed operators can be applied to study the impulsive difference equations and their dynamics. Additionally, a feed-forward ANN model is established to approximate the analytic expressions involved in inequalities and to observe the consistency of integral bounds. The results of the ANN analysis suggest a significant agreement between analytical solutions and approximations. Lastly, we focus on an applicable analysis of our derived results. The generic nature of operators will lead to new developments in quantum calculus. The hybrid approach evolved in this study will bring new applicable insights to the mathematical analysis. Full article

Pseudocereals such as quinoa, amaranth, and buckwheat have attracted increasing attention as ingredients for extruded snack products because of their nutritional value, gluten-free status, and content of bioactive compounds. The quality of extruded products is governed by complex interactions among processing variables, including barrel temperature, screw speed, feed moisture content, and formulation characteristics. As a result, mathematical modelling has become an important tool for predicting product properties and identifying suitable processing conditions. This review summarizes modelling approaches applied to extruded food products with a focus on pseudocereal extrusion. Particular emphasis is placed on response surface methodology (RSM), artificial neural networks (ANNs), adaptive neuro-fuzzy inference systems (ANFIS), support vector regression (SVR), and hybrid optimisation strategies. Published studies indicate that RSM remains the most commonly used approach because of its simplicity and interpretability, while ANN-based models generally provide much higher predictive accuracy when strong nonlinear relationships are present. The widespread use of small experimental datasets and limited external validation remains a major challenge for the practical implementation of advanced machine-learning models. This review examines the strengths and limitations of current modelling approaches and discusses future opportunities for integrating predictive models with digital manufacturing frameworks. Full article

Broadband, low-frequency sound absorption within a compact device remains a central unsolved problem in noise control engineering, arising from fundamental trade-offs among resonator volume, absorption bandwidth, panel thickness, and frequency tunability that no passive, linear, time-invariant system can simultaneously circumvent. This review establishes a unified mass–spring–damping (MSD) framework applied systematically across the full spectrum of resonance-based absorber technologies. From first principles, we derive the mass–stiffness coupling result (the mass-disappearing result of Shen and Liu): fixing the resonance frequency imposes $K=M{\omega }_{\mathrm{res}}^2$, so acoustic mass and stiffness cannot be adjusted independently; the half-absorption bandwidth ${\Pi }_1=\eta /\left(M{\omega }_{\mathrm{res}}\right)+V{\omega }_{\mathrm{res}}/\left(c_0{S}_{\mathrm{tar}}\right)$ then depends explicitly on the cavity volume V (system stiffness) and on the damping coefficient $\eta$, rather than on mass as an independent lever. This explains why neck extension, space-coiling, and membrane loading—which merely add mass while leaving the cavity stiffness unchanged—fail to broaden the absorption band at fixed volume, and refocuses the design effort on stiffness reduction and damping control. Five non-dimensional performance metrics are introduced that collapse the scattered literature into a single, scale-independent language for rigorous comparison across all absorber families: normalised half-absorption bandwidth ${\Pi }_1$, volume efficiency ${\Pi }_2$, integral absorption criterion ${\Pi }_3$ tied to the Rozanov causality bound, quality factor $Q=1/{\Pi }_1$, and frequency-thickness ratio ${\Pi }_4$. A two-degree-of-freedom acoustic–structural coupling model yields closed-form effective stiffness and damping, revealing how structural loss augments acoustic damping, how modal veering produces split absorption peaks, and how the anti-resonance frequency becomes a designable parameter. A critical distinction is drawn between mathematical negative stiffness (a fitting artefact) and physical negative stiffness via repulsive magnets, bistable elements, or negative-capacitance piezoelectric shunts, which genuinely reduces cavity stiffness, lowers resonance frequency, and widens bandwidth beyond the passive causality bound. The shunt electromechanical diaphragm further demonstrates $\alpha >0.9$ at nine tonal frequencies spanning three octaves without mechanical modification. Finally, embedding MSD equations and ${\Pi }_1$–${\Pi }_4$ bounds as hard physical priors in AI/LLM-assisted design frameworks is identified as the key step toward provably physically consistent absorber synthesis. Full article

Backpropagation is less a single algorithm than a pipeline of choices: how the error signal is propagated, how the weight gradient is assembled, and how the update is applied. This paper revisits three consecutive steps and proposes small, mathematically transparent modifications that improve gradient scaling and conditioning without changing the represented function class. The resulting method, Conditioned Backpropagation(CBP), combines (i) a layerwise gradient-norm equalization that counters the geometric depth dependence of the backpropagated error; (ii) an activation-centering reparameterization that removes the dominant rank-one mean term from the per-layer curvature; and (iii) a damped diagonal preconditioner that is positive-definite by construction. The composite operator is a bounded positive-definite preconditioner, so the method inherits standard nonconvex, Polyak–Łojasiewicz, and stochastic convergence guarantees at the per-step cost of ordinary backpropagation. No prior method composes these three repairs into one operator with a joint boundedness and positive-definiteness guarantee. Two further results, both new, concern equalization. On a block-structured strongly convex model, and for the curvature-equalizing target that the implemented gradient-energy equalizer approximates up to a quantified heterogeneity factor, equalization makes the convergence rate depth-uniform; the bounded-clip version that is actually run stays depth-uniform up to a clip-determined depth and retains a constant-factor improvement beyond it. Controlled experiments, run over ten or more seeds with paired significance tests, confirm the mechanisms: Equalization compresses an order-of-magnitude per-layer gradient disparity, centering cuts the top curvature eigenvalue about threefold and yields the lowest training loss, and the configurations combining centering with the damped preconditioner, including the full method, converge fastest. The effects persist on MNIST and on CIFAR-10 with a small residual convolutional network, at a measured per-iteration overhead below about twice that of Adam. Generalization is comparable across methods, and no end-to-end depth-scaling advantage is claimed, keeping the contribution focused on optimization geometry. Full article

The Rayleigh–Logarithmic distribution provides a useful framework for modelling lifetime data by combining continuous lifetime variability with a logarithmic compounding mechanism. This study introduces a three-parameter Alpha Power Rayleigh–Logarithmic (APRL) distribution by applying the Alpha Power transformation to the classical Rayleigh–Logarithmic model. The additional transformation parameter allows the distributional shape, skewness, tail behaviour, and rate of increase in the hazard function to be adjusted while retaining the underlying structure of the baseline model. Several mathematical and reliability properties of the APRL distribution are derived, including the probability density and cumulative distribution functions, survival and hazard rate functions, quantile function, moments, order statistics, and mean residual life function. Model parameters are estimated by maximum likelihood using a multiple-start numerical optimization procedure, and the finite-sample performance of the estimators is investigated through Monte Carlo simulations under different parameter configurations and sample sizes. The simulation results show that estimation accuracy generally improves with increasing sample size, as reflected by decreasing bias, MSE, and RMSE, although estimation of the transformation parameter may exhibit greater variability for more extreme parameter settings. The practical performance of the APRL distribution is examined using the Aircraft Windshield Failure Times and Breaking Stress of Carbon Fibres datasets. Model comparisons based on information criteria, bootstrap-based goodness-of-fit assessment, and graphical diagnostics show that the APRL distribution provides competitive fits relative to several established lifetime distributions. In addition, mean time to failure and mean residual life analyses illustrate the practical interpretation of the reliability measures derived for the proposed model. Overall, the results support the APRL distribution as a useful alternative for the statistical analysis of lifetime and reliability data. Full article