This research presents an analytical examination of the steady-state boundary layer flow and thermal characteristics of hybrid nanofluidsystems. Both nanofluid and hybrid formulations are analysed; a similarity transformation recasts governing partial differential equationsinto ordinary differential forms. The derived system is tackled using the Homotopy Asymptotic Method (HAM), which provides closed-form analytical solutions that contribute to an understanding of the flow behaviour. The researcher produces visual representations of the flow and temperature fields for the different cases and performs a convergence analysis using the BVPh 2.0 software, which involves 25 iterative cycles. The skin friction coefficient and Nusselt number are two key parameters. The results serve as a fundamental step for engineering applications of advanced fluid mechanics, which include the manufacture of materials, the transport of biomedicals, cooling systems, and thermal management.

Combinatorial differential forms for multi-dimensional fluid flow in porous media: A unified framework for volumetric pores, fractures, and channels

ABSTRACT This study examines the impact of electromagnetic Lorentz force and thermal convective boundaries on second‐order nanofluid flow over a Riga plate, considering heat sources varying with both location and temperature. A Lorentz force acts along the plate, and the Grinberg‐related component is included in the governing equations. A no‐mass‐transfer condition on the solid Riga boundary suppresses nanoparticle concentration. Using modern heat and mass transfer principles, the Cattaneo–Christov thermal flux model and extended Fick's law are applied. The governing equations are transformed into ordinary differential form through similarity transformations and solved using the Runge–Kutta–Fehlberg method. Results show enhanced nanoparticle temperature with higher radiation parameters and Biot numbers, whereas concentration decreases under constructive and increases under destructive chemical reactions.

Abstract Given a Hermitian holomorphic vector bundle over a complex manifold, consider its flag bundles with the associated universal vector bundles endowed with the induced metrics. We prove that the universal formula for the push‐forward of a polynomial in the Chern classes of all the possible universal vector bundles also holds pointwise at the level of Chern forms. A key step in our proof is the explicit computation, at a point of any flag bundle, of the Chern curvature of the universal vector bundles with the induced metrics. As an application, we provide an alternative version of the Jacobi–Trudi identity at the level of differential forms. We also show the positivity of a family of polynomials in the Chern forms of Griffiths semipositive vector bundles. This latter result partially confirms the Griffiths' conjecture on positive characteristic forms, which has raised considerable interest in recent years.

This study introduces a hybrid numerical-analytical methodology for interpreting, for the first time, both the amplitude and phase of Synchrotron Infrared Nanospectroscopy (SINS) spectra of hydroxyapatite (HA) thin films. The approach combines the Self-Referenced Interferometry Model (SRIM) with the Kramers-Kronig relations (KKR). HA films with thicknesses of 100 nm and 600 nm were fabricated via magnetron sputtering and characterized using Fourier-transform infrared (FTIR) spectroscopy to identify vibrational modes of the phosphate group. These frequencies were used as initial parameters for Lorentzian-based SRIM fitting of the SINS amplitude spectrum. Both the integral and differential forms of KKR were applied to the fitted amplitude spectrum to reconstruct the phase, which was subsequently validated against experimental SINS data. The differential form further enabled analytical decomposition of the phase into individual oscillator and non-resonant contributions, revealing features not evident in the amplitude alone. The strong agreement between experimental and modeled spectra demonstrates the reliability of FTIR-guided SRIM fitting combined with KKR analysis. This methodology effectively bridges far-field and near-field infrared spectroscopy, showing that HA spectral responses comply with causality and analyticity. Moreover, it supports accurate phase reconstruction even under low-intensity signal conditions, indicating broad applicability to other nanostructured materials. Altogether, this work establishes a robust framework for interpreting SINS spectra and advances phase-resolved diagnostics and hybrid modeling strategies for nanoscale vibrational analysis.

Integration in non-integer-dimensional spaces (NIDS) is actively used in quantum field theory, statistical physics, and fractal media physics. The integration over the entire momentum space with non-integer dimensions was first proposed by Wilson in 1973 for dimensional regularization in quantum field theory. However, self-consistent calculus of integrals and derivatives in NIDS and the vector calculus in NIDS, including the fundamental theorems of these calculi, have not yet been explicitly formulated. The construction of precisely such self-consistent calculus is the purpose of this article. The integral and differential operators in NIDS are defined by using the generalization of the Wilson approach, product measure, and metric approaches. To derive the self-consistent formulation of the NIDS calculus, we proposed some principles of correspondence and self-consistency of NIDS integration and differentiation. In this paper, the basic properties of these operators are described and proved. It is proved that the proposed operators satisfy the NIDS generalizations of the first and second fundamental theorems of standard calculus; therefore, these NIDS operators form a calculus. The NIDS derivative satisfies the standard Leibniz rule; therefore, these derivatives are integer-order operators. The calculation of the NIDS integral over the ball region in NIDS gives the well-known equation of the volume of a non-integer dimension ball with arbitrary positive dimension. The volume, surface, and line integrals in D-dimensional spaces are defined, and basic properties are described. The NIDS generalization of the standard vector differential operators (gradient, divergence, and curl) and integral operators (the line and surface integrals of vector fields) are proposed. The NIDS generalizations of the standard gradient theorem, the divergence theorem (the Gauss–Ostrogradsky theorem), and the Stokes theorem are proved. Some basic elements of the calculus of differential forms in NIDS are also proposed. The proposed NIDS calculus can be used, for example, to describe fractal media and the fractal distribution of matter in the framework of continuum models by using the concept of the density of states.

Abstract Weights are geometrical degrees of freedom that allow to generalize Lagrangian finite elements. They are defined through integrals over specific supports, well understood in terms of differential forms and integration, and lie within the framework of finite element exterior calculus. We adopt this formalism with the target of identifying supports that are appealing for a finite element approximation, describing weights in terms of a single parameter in the third- and fourth-order methods. To do so, we study the related parametric matrix-sequences, with the matrix order tending to infinity as the mesh size tends to zero. Using the generalized locally Toeplitz theory, we analyze the performance of weights-based finite elements on an elliptic operator. In particular, for degrees 3 and 4, we identify an optimal value for the weights location, which sits in a rather large interval where weights give rise to better conditioned stiffness matrices. With this at hand, we propose and test ad hoc preconditioners, in dependence of the discretization parameters and in connection with conjugate gradient method. The model problem we consider is a one-dimensional Laplacian, both with constant and non-constant coefficients. Numerical visualizations and experimental tests are reported and critically discussed, showing a confidence interval for the choice of the parameter.

The subject of the research is a mathematical model of the equilibrium of a solid body of satisfactory spatial shape under external load. The development of solid mechanics is largely related to practical goals — calculations of the strength of structural elements and machine parts, the violation of which is usually understood as reaching a state where the structural properties of the product change, rendering it unsuitable for use. Many years of research into the system of differential equations in partial derivatives of the laws of equilibrium in mechanics, known as the Navier-Lame system of equations, has not yet, due to its complexity, achieved the level of development that would guarantee the possibility of obtaining analytical solutions. This raises many questions and misunderstandings regarding the solutions of boundary value problems using widely available application packages based on finite difference approaches. The purpose of this article is to present the results of the development of an alternative method of boundary integral equations, which, due to the boundary conditions of deformation of a continuous medium, leads to a system of linear boundary integral equations with the existing set of solutions. Task: to construct integral representations of solutions to the system of differential equations of equilibrium laws of a solid deformable body using the generalized potential theory method for differential operator forms of the corresponding equations. Scientific novelty. The differential operations of vector-tensor analysis are developed and generalized. The generalized integral theorems for differential operators of the second order adequate to the equilibrium laws of a deformable solid are proved. The results obtained. On the basis of the created generalized apparatus of vector-tensor analysis, analytical solutions in the form of integral representations of the main kinematic characteristics of the problem of deformation of a solid of satisfactory spatial form are constructed. Conclusions. The boundary value problem of the equilibrium of a deformed solid in the presence of an external force is reduced to a system of linear boundary integral equations with respect to the kinematic characteristics of the problem. In addition, it is proved for the first time that all the obtained characteristics are related to the vector momentum potential, which greatly simplifies the integral representation of solutions and numerical implementation of solutions of the corresponding integral equations.

Quantum systems in the hyperbolic phase-space: Explicit maps, differential form of the star product and their applications

ABSTRACT This study presents an in‐depth analysis of mixed convection flow through a porous medium incorporating the effects of internal heat generation and heat absorption. The novelty of this study lies in the combined evaluation of localized heat sources, chemical reactivity, and magnetohydrodynamic (MHD) influences within a single porous configuration—an approach rarely addressed collectively in prior work. The governing equations, expressed as nonlinear, dimensionless ordinary differential forms, are solved using the regular perturbation method. The results reveal the combined influence of these factors on velocity, temperature, and concentration profiles, offering insights applicable to engineering applications such as advanced cooling systems, filtration processes, fuel cell design, and biomedical devices like blood purification units.

This study analyzes the burned area and fire severity of the forest fires that occurred in Ourense in August 2025. The analysis was based on Sentinel-2 imagery, which provides near-infrared (NIR) and short-wave infrared (SWIR) bands suitable for distinguishing between burned and unburned surfaces, as well as different severity levels. Several spectral indices widely applied in wildfire remote sensing were employed, including the Normalized Burn Ratio (NBR), the Burned Area Index for Sentinel-2 (BAIS2), and the Relativized Burn Ratio (RBR), together with their differential forms (e.g., dNBR). Temporal comparisons between pre- and post-fire conditions allowed a multitemporal assessment of fire progression and impact. Field plots were compared with the satellite-derived classification to evaluate accuracy (88%). The results demonstrate that Sentinel-2 data enable a precise delineation of burned areas and a reliable classification of severity levels in forest fires in Ourense. This study confirms the potential of satellite-based indices, supported by limited field validation, to provide rapid and accurate information for forest fires assessment and post-fire management.

Purpose Harnessing nanofluids in the cone–disk model drives advanced cooling systems, heat exchangers and thermal management, transforming aerospace, electronics and automotive sectors. Its rotational dynamics and confined geometry make it ideal for microelectromechanical systems, rotary sensors and compact cooling devices requiring precise heat control. The exceptional thermal conductivity and adjustable flow properties of nanofluids enable superior temperature regulation, particularly in high-speed, confined environments. Design/methodology/approach Numerical solutions to the intricate governing partial differential equations are derived using the MATLAB-based bvp4c algorithm, with equations reduced to ordinary differential form via a similarity transformation. Findings The contribution of key parameters to flow dynamics and thermal performance is comprehensively presented through both graphical and tabular formats. The key findings reveal that smaller nanoparticle radius and larger interparticle spacing enhance the flow behavior. Observations indicate that, in most cases, the Nusselt number on the cone surface surpasses that of the disk. Originality/value To the best of author’s knowledge, no study in literature exists that discusses the ramifications of nanoparticle radius and interparticle spacing on the dynamics of nanofluid flow within a cone–disk apparatus, using MoS2 nanoparticles dispersed in kerosene oil. Additionally, this study explores the fluid’s non-Newtonian rheology, articulated through a tangent hyperbolic model, while accounting for the impact of irregular heat sources and power index on the system’s thermo-fluidic phenomena.

Inequalities between Dirichlet and Neumann eigenvalues of the Laplacian and of other differential operators have been intensively studied in the past decades. The aim of this paper is to introduce differential forms and the de Rham complex in the study of such inequalities. We show how differential forms lie hidden at the heart of the work of Rohleder on inequalities between Dirichlet and Neumann eigenvalues for the Laplacian on planar domains. Moreover, we extend the ideas of Rohleder to a new proof of Friedlander’s inequality for any bounded Lipschitz domain.

The design of a flapping-wing flying robot (FWFR) requires lightweight components to increase the thrust-to-weight ratio for the robot; therefore, the components are subjected to a trade-off between strength, power, and weight. The processor of the FWFR must be selected as lightweight as possible, and satisfy the computational speed requirements. Microcomputers are suitable, subject to the use of a simple operating system and programme for control implementation; this approach avoids massive computations. In the execution of nonlinear optimal control, the state-dependent Riccati equation (SDRE), on such a setup, predefined libraries should be used, which impose a delay in the solution. A solution method to the SDRE is suggested, considering the differential form of the Riccati equation, the so-called state-dependent differential Riccati equation (SDDRE). A solution using forward integration is introduced to increase the sampling rate significantly. This faster solution provides the possibility of a safer control implementation.

The logistic modeling of diauxic growth and biphasic antibacterial activity (AA) production was enhanced for four lactic acid bacteria (Lactococcus lactis CECT 539, Pediococcus acidilactici NRRL B-5627, Lactobacillus casei CECT 4043, and Enterococcus faecium CECT 410) during realkalized fed-batch fermentations. The improved growth model, also validated for describing the diauxic growth of Mos breed roosters and foals, overcomes a key limitation of the bi-logistic model, which assumes the existence of two distinct populations growing from the start of the culture, each following a different growth profile. In contrast, the improved logistic growth model developed in this study accounts for a single population growing at two rates, offering a fit to the experimental data comparable to that of the commonly used bi-logistic model. The enhanced model for product synthesis accurately describes biphasic AA production, assuming that antibacterial products are synthesized as growth-associated metabolites, depending on the final pH reached in the cultures at each sampling time. Additionally, it is easier to apply than the unmodified or modified differential forms of the Luedeking–Piret model. This study demonstrated, for the first time, the applicability of these two models in describing the diauxic growth and biphasic AA synthesis of LAB.

A weighted Reilly formula for differential forms and sharp Steklov eigenvalue estimates

Abstract We consider the Steklov problem on differential $$p\text {-}$$ p - forms defined by Karpukhin and present geometric eigenvalue bounds in the setting of warped product manifolds in various scenarios. In particular, we obtain Escobar type lower bounds for warped product manifolds with non-negative Ricci curvature and strictly convex boundary, and certain sharp bounds for hypersurfaces of revolution, among others. We compare and contrast the behaviour with known results in the case of functions (i.e., $$0\text {-}$$ 0 - forms), highlighting the influence of the underlying topology on the spectrum for $$p\text {-}$$ p - forms in general.

Abstract The study of tangent hyperbolic nanofluids in the presence of bioconvection and nonlinear thermal radiation over Riga plates within porous media addresses critical challenges in enhancing heat transfer and fluid dynamics in advanced engineering systems. This research fills a significant gap by applying artificial neural networks (ANNs) to model the complex behavior of tangent hyperbolic nanofluids, which exhibit non‐Newtonian characteristics, under these conditions. By introducing appropriate similarity transformations, the governing equations in partial differential form are reduced to ordinary differential equations. These resulting equations are then integrated numerically using the Runge‐Kutta method of order four. This innovative approach offers both precision and computational efficiency in addressing highly nonlinear systems. The findings have substantial real‐world applications, particularly in the optimization of heat transfer technologies, such as thermal management systems, bio‐microfluidic devices, and enhanced oil recovery in porous media. The integration of ANNs with classical numerical techniques provides a robust framework for solving fluid flow problems in complex environments, offering new avenues for improving energy systems and industrial processes where precise thermal control is crucial.

On the truth of integral and differential constitutive forms in strain-driven nonlocal theories with bi-Helmholtz kernels for nanobeam analysis

In this work, a new NURBS-based isogeometric finite element formulation is developed to study the free vibration responses of complex axially functionally graded (AFG) nanobeams using the stress-driven two-phase local/nonlocal integral model. The material composition varies according to a power-law distribution in the axial direction. The governing equations and standard boundary conditions are derived using Hamilton’s principle. The integral constitutive equations are equivalently converted into differential forms, accompanied by the corresponding constitutive boundary conditions. A finite element method (FEM)-based isogeometric analysis (IGA) is developed for the vibration responses of AFG nanobeams under different boundary conditions. With the aid of the constitutive boundary conditions, the model can flexibly accommodate higher-order variables. The deflection values can be modeled by NURBS (Non-Uniform Rational B-splines), allowing for the implementation of arbitrarily functionally graded microstructures via IGA. The results of the present AFG nanobeam model are compared with those obtained using DQEM (generalized differential quadrature method) to demonstrate the validity of the isogeometric model. Numerical results highlight the significance of the nonlocal parameter, the AFG index, and the length-to-height ratio on the vibration behavior.