Vol. 14, No. 3 (2025): Regular Issue

Based on the mutual relations among the well-known Banach contraction mapping (1922), Rakotch contraction mapping (1962), Kannan contraction mapping (1968), Bryant contraction mapping (1968), and Reich contraction mapping (1971) - particularly considering corresponding examples, purposeful and deliberate counterexamples, and the remarkable analytic properties of the Bryant contraction mapping, which is the most explicit and simple generalisation of the Banach contraction mapping - we confirm, for the first time, that only a few contractive mappings, such as the Rakotch contraction mapping, can generate fractals. At the same time, we establish that one cannot always generate fractals using various generalised Banach contraction mappings in a standard Euclidean metric space, as most of the analytic properties of these contraction mappings on the base space are not necessarily inherited by the mappings they generate on the fractal space.

This work considers the flow past a horizontally moving needle submerged in Casson fluid. The viscosity and thermal conductivity are assumed to be dependent on temperature. The flow-governing equations are changed into a set of non-linear ordinary differential equations using appropriate transforms. Applying successive linearization, the resulting equations are linearized, and then the Chebyshev spectral collocation technique is implemented. The effects of the Casson fluid parameter, needle size, and viscosity parameter on velocity and temperature, along with graphical representations of the coefficient of skin friction and local heat transfer rate, are presented graphically. It is observed that an enhancement in the viscosity parameter results in a decrease in the heat transfer rate and an increase in velocity. Temperature, velocity, and heat transfer rate all increase as the thermal conductivity parameter increases.

This investigation is made to study the effect of exponentially increasing wave amplitude on swallowing of such food stuffs which behave as Casson fluid. Highly concentrated fluids such as tomato puree, jelly, soup, honey etc has been considered. The dependence of pressure on space and time has been investigated for time averaged flow rate. The magnitude of the pressure along the length of oesophagus increases with the larger wave amplitude amplifying parameter at all temporal values. It is also observed that the difference between the maximum and the minimum pressures at the distal end becomes greater when amplitude increases exponentially.

There are a large number of complex nonlinear behaviors, such as separation bubble and mass transport, in the unsteady flow separation and complicated flows around airfoils at typical stalls under low Reynolds number, and the explanations for these nonlinear behaviors are still open problems. At present, the research on unsteady separated flow is mainly based on the Eulerian description. However, the descriptions and definitions of some complex unsteady flow phenomena in complicated separation flow and others are somewhat not general, because of the lack of temporal correlation and dynamic property in Eulerian description, and such situations have further influences on the flow control, as one of the current challenges in aerodynamics. In contrast, Lagrangian description methods are objective and could accurately describe and analyze the nonlinear dynamical behaviors, such as mass transport and mixture in flows. Therefore, an efficient Lagrangian analysis method is introduced in this work, to study the unsteady flow separation and the complicated flow around airfoil at typical stalls under low Reynolds number, with combination of theoretical analysis and numerical method. In particular, the evolution of separation bubble and the mass transport in complex unsteady flow separation can be captured and revealed by tracing the intrinsic Lagrangian coherent structures, and the controlling of unsteady flow separation can be analyzed in depth, with unsteady excitations. First of all, an efficient Lagrangian analysis method, combining the method of grid partition matching and automatic adjustment of integral time, is introduced to analyze the dynamic behaviors and to save computing time, due to that a large number of fluid particles need to be tracked in the process of Lagrangian coherent structures (LCSs) calculation. Then, the evolution of LCSs near the airfoil at low Reynolds number is analyzed in detail, using LCSs and Lobe dynamics based on nonlinear dynamics. In particular, the mass transport and transport channels between the flow separation bubble on the airfoil surface and the mainstream are described dynamically, and the relationship between the mass transport and the lift is investigated further. In addition, by comparing the intrinsic LCSs and mass transports in the flow around airfoil at stall and the airfoil controlled by synthetic jet, it can be concluded that the unsteady excitation could modify or adjust the topology of flow structures near the airfoil, and then induces the mass transport to improve the aerodynamic performances. Furthermore, the flow control with unsteady excitation on the post-stall airfoil is studied numerically and theoretically, and the influences of the amplitude and frequency of unsteady excitation on the mass transport, in a targeted transport way, are explained with LCSs in depth. It is found that there are mainly four routes in mass transport to enhance the lift of airfoil at post-stall attack angle using synthetic jet, and the jet parameters modify or adjust the aerodynamic performances through the four routes. In summary, with introduction of an efficient numerical analysis method of Lagrangian coherent structures and Lobe dynamics, the evolutions of flow structures, mass transports in separation flow and typical stalls of airfoils at low Reynolds number are studied in detail, gaining some key understandings and providing theoretical explanations for the flow separation and typical stalls of airfoils. Also, one reliable theoretical basis is paved for improving aerodynamic performances of airfoil using unsteady excitation.

In this article, we study the quasigeostrophic equation, which is a prototypical geophysical fluid model. We will show the existence of positive solutions and almost time-periodic solutions.

This paper deals with the linear matrix inequality (LMI) conditions for output feedback control problems defined by continuous matrix systems. The solution of nonlinear matrix inequality obtained in the Lyapunov function approach for the stabilization of matrix Lyapunov and Sylvester systems is obtained in terms of the solutions of corresponding linear matrix inequalities (LMIs). The established results are based on sufficient conditions since they are dependent on the state-space representation used for describing the continuous time matrix Lyapunov and Sylvester systems. Linear Matrix Inequalities (LMI) are constructed to establish sufficient conditions for the continuous time matrix Lyapunov and Sylvester systems. The established conditions demonstrated by a numerical examples.

This paper introduces a method for correcting wavefront aberrations using intensity images obtained under varying phase diversities. The proposed methodology integrates Gaussian mixture model (GMM) clustering and employs the resulting cluster centers as initial positions for a Limited memory Broyden Fletcher Goldfarb Shanno (L-BFGS) optimization process. The study evaluates the performance of this method using simulated data. The dataset consists of 500 distinct aberrations characterized by root mean square (RMS) errors ranging from 0.2$\lambda$ to 0.3$\lambda$. The analysis focuses on assessing the accuracy achieved, particularly emphasizing the effectiveness of wavefront reconstruction. The obtained RMS residual errors range from 0.017$\lambda$ (lowest) to 0.066$\lambda$ (highest), with an average of 0.039$\lambda$. Notably, 89.6% of the RMS residual errors fall below 0.05$\lambda$. These results demonstrate the reliability and practicality of the proposed approach in wavefront aberration correction, as confirmed through numerical experimentation. Results can improve applications requiring precise wavefront correction for clear and detailed observations, such as astronomical imaging or high-resolution microscopy.

In the present work, a model is proposed where the ferroptosis process is related to the stages of avascular and vascular tumor growth, including the process of epithelial-mesenchymal transition. It was found that the ferroptosis process induces cell death more effectively in cells with a mesenchymal phenotype. Furthermore, it was observed that for certain compositions of epithelial and mesenchymal cells the tumor growth system exhibits less complexity. From the evaluation of the entropy production rate, it was identified that an increase in the concentration of oxidized lipid peroxide species generates an increase in the robustness of the ferroptosis process, while decreasing the robustness of the epithelial-mesenchymal transition mechanism.

We formulate nonlinear nonlocal integro-PDE with memory, biloaded (boundary integrals load the ambient space, and the ambient space loads the boundary), and the associated optimal control problems. We derive part of the necessary conditions for optimality in the form of Hamilton-Euler-Lagrange loaded integro-PDEs. In the process, we introduce an agglomeration of new differential operators. Our results have relevance to optimal amelioration of flooded areas, remediation of sites of contaminated groundwater, and active control methods for optimally extinguishing forest fires.

Fractal interpolation offers a more flexible approach than traditional interpolation methods, thanks to techniques that use repeated transformations to build these unique functions. Fixed points -- the values that remain stable through these transformations -- are central to creating patterns that stay consistent, or `invariant', across different types of iterated function systems. Unlike classic methods like polynomial interpolation, fractal interpolation achieves its shape through a specialised operator, which ensures that the function holds its fractal form. A common foundation for this approach is the Banach fixed point theorem, which helps in reliably constructing these functions. This article reviews key methods for creating 2D fractal surfaces, focusing on techniques like Rakotch contractions and the Matkowski theorem, which expand the possibilities for fractal interpolation in real-world applications. It also includes some of the authors' recent findings. Notably, we build on past work with bivariable fractal interpolation functions to offer a more detailed perspective. The methods presented here diverge from existing techniques, offering straightforward ways to represent complicated patterns.

We consider and apply a multidimensional discrete-time delay autonomous third order non-linear vector difference equation system, where the orthogonal change after two reflections is given by a vector cross product leading to spinor rotations $X_{i}-X_{i-2} = -X_{i-1}\times F_{i}\left(X_{i-1},X_{i-2},X_{i-3}\right)\in\mathbb{R}^{3}\textrm{ or}\in\mathbb{R}^{7}$, where the symmetry invariant $C=X_{i}\cdot X_{i-1}=X_{i-1}\cdot X_{i-2}=...$ allows at every step for a memory sign flip including expansion/contraction by a scalar factor. Using the Frenet frame approach defining orthogonal co-moving components with torsion and curvature parameter, both, the orthogonal frame and the change of the frame are represented by the three position memory terms recurrently. The necessary cross and dot vector product (split-) algebra is encoded in variable multiplication tables in 3d and 7d. We discuss two special $F_{i}$ types showing stable point densities, which are drifting limit cycles with sub-cycles, where the resulting smooth orbital spinor dynamics shows discrete atomic-type orbital eigenstates or local waves with characteristic numbers, non-local reflection, instability, hysteresis, interaction, helical emissions, and transition to chaos.

In this work, we analyze the evolution of the phase-space structures of KCN molecular system as a function of the vibrational energy using Lagrangian descriptors. For low energies, the motion is mostly regular around the absolute minimum of the potential energy surface. As the energy increases, the phase space combines regions with regular and chaotic motion, a difference that is well captured by the Lagrangian descriptors. We show that the dynamics is mostly governed by the invariant manifolds of the stretch periodic orbits located at the top of one of the energetic barriers of the system. Furthermore, we show a perfect agreement between the bifurcation theory and the differences observed in the phase-space structures as the vibrational energy is modified. The accuracy of our calculations is also assessed by explicit comparison with the invariant manifolds computed using linear dynamics.