In this work, we introduce and investigate a novel class of polynomials, which we term t-successive exponential partial Bell and r-Bell polynomials. These polynomials generalize the well-studied exponential partial Bell and r-Bell polynomials and are closely related to classical combinatorial sequences such as Stirling numbers, Bell numbers, and their polynomial extensions. Our motivation stems from the unified framework proposed by Mihoubi and Rahmani in(2017), which provides a systematic approach for constructing exponential partial r-Bell polynomials, of which many known combinatorial numbers emerge as special cases.

We begin by establishing explicit formulas for these polynomials and presenting their generating functions, which serve as a fundamental tool for deriving structural properties. Through careful analysis, we deduce several recurrence relations that encapsulate the inherent combinatorial structure of these polynomials. In particular, we provide combinatorial interpretations in terms of set partitions, illustrating how elements may be organized into subsets under specific constraints, thereby extending the classical interpretations associated with Bell and Stirling numbers.

t-successive exponential partial Bell and r-Bell polynomials exhibit rich combinatorial and algebraic properties. They offer a unifying perspective that bridges existing sequences and opens new avenues for exploration in discrete mathematics and combinatorial enumeration. We illustrate their utility through representative examples and demonstrate how their generating functions facilitate the derivation of identities and recurrence relations.

Our study not only generalizes known results but also provides a versatile framework for further investigations. By combining explicit constructions, recurrence relations, and combinatorial interpretations, these polynomials contribute to a deeper understanding of the interplay between classical combinatorial sequences and their extensions. The results presented herein offer potential applications in sequence transformations, partition theory, and the broader study of combinatorial structures.

Ordinary differential equations with nonlocal and integral boundary conditions have attracted significant attention due to their theoretical importance and wide range of applications. In this work, we investigate a spectral problem for an ordinary differential operator subject to integral conditions involving both the first and second derivatives of the unknown function.

The novelty of this work lies in the formulation of more general integral conditions that extend those studied by K. A. Darovskaya and A. L. Skubachevskii, where the integral conditions involve either the first or the second derivative of the unknown function. The proposed framework allows for a unified treatment of such conditions and leads to a broader class of differential operators.

A priori estimates for the solutions of the problem are obtained for sufficiently large values of the spectral parameter λ. These estimates play a key role in the analysis and provide a basis for studying the corresponding operator. In particular, they are used to investigate the spectral properties of the associated operator and to ensure the well-posedness of the problem.

The approach is based on methods of functional analysis and the theory of Fredholm operators. The obtained results generalize several known results in the literature and contribute to the development of the theory of differential equations with nonlocal conditions.

In this work we studied a nonlinear dynamic systems model that helps us explain the complicated behavior of financial markets. This model shows how the prices of financial assets affect each other and investors’ decisions over time, creating feedback loops. Some of these loops can help keep the market stable, while others may make it more unstable.

First, we will check the well-posedness of this model to make sure that all the equations used give meaningful results when tested under realistic conditions. After confirming that our theoretical model works properly, we will study how stable the model is and find the conditions that help the market stay balanced as well as the conditions that can cause bigger swings.

Regulatory policies, intervention rules, or automated market systems will also be part of applying the theoretical study. The main goal of these types of feedback control mechanisms is to improve market stability and prevent extreme price changes; therefore, their use in crises could be helpful for preventing problems in the future.

We will also run many numerical simulations to show how the model behaves under different situations and control strategies. This will give useful insights into financial market behavior and show how mathematical modeling can help the financial industry. Overall, this study helps explain market dynamics and offers a practical way to help institutions understand and predict future market situation.

Classical numerical approaches to Fredholm integral equations often rely on an implicit orthodoxy: discretise first, iterate afterwards. While effective in numerous settings, this paradigm becomes less transparent when the equation is defined on large intervals, where the infinite-dimensional structure of the problem quickly perturbs finite-dimensional approximations and obscures much of the anticipated benefit of classical schemes.

In this work, we recalibrate the compass by exploring a reversed paradigm. Rather than allowing discretisation to dominate the analysis from the outset, we begin by reformulating the equation as a system of coupled operator equations acting on a product Banach space, induced by a subdivision of the underlying domain. This naturally leads to an operator matrix whose entries are bounded linear operators, allowing notions familiar from linear algebra to be revisited within an infinite-dimensional framework.

Within this setting, we introduce a generalised successive over-relaxation (GSOR) scheme tailored to such operator matrices. A central theoretical result establishes that, under suitable conditions, the operator matrix is strictly diagonally dominant with respect to the operator norm. This structural property governs the convergence analysis of the method: it ensures the invertibility of the operator system and yields convergence of the GSOR iteration in the Banach space setting.

Rather than being merely a numerical device, our contribution clarifies how classical relaxation ideas can migrate, with minimal distortion, from finite-dimensional linear systems to systems of bounded linear operators. This perspective highlights a genuine theoretical bridge between linear algebra and functional analysis and suggests that iterating at the operator level offers a structurally sound and conceptually natural way to approach Fredholm integral equations on large intervals.

In this work, we investigate the qualitative dynamics of a nonautonomous predator–prey model incorporating a saturated Holling-type functional response and an additive Allee effect in the prey population. Such a framework is biologically relevant since ecological parameters often fluctuate in time due to seasonal or environmental variations, while Allee mechanisms may strongly influence prey growth at low population densities.

First, by applying the comparison principle and constructing suitable differential inequalities, we establish sufficient conditions ensuring the positivity and permanence of the system. In particular, we prove that both prey and predator populations remain uniformly bounded
away from extinction whenever appropriate persistence thresholds are satisfied. Next, a Lyapunov-type function based on logarithmic distance is introduced in order to derive explicit criteria for the global stability of positive solutions. This analysis guarantees that all trajectories starting from positive initial values asymptotically approach each other, showing that the system exhibits robust long-term behavior under the permanence regime.

Furthermore, we obtain threshold conditions under which the predator population becomes extinct asymptotically, revealing how the combined effects of mortality, predation saturation, and the Allee mechanism can lead to predator disappearance even when prey survive.

In addition to the theoretical results, numerical simulations are performed to illustrate the validity of the analytical findings and to explore the richness of the model’s dynamics. In particular, for certain parameter regimes, especially in the region of weak Allee effect, the system exhibits chaotic dynamics characterized by irregular oscillations, sensitive dependence on initial conditions, and complex attractors.

Mathematical modeling of tumor–immune dynamics has become an essential tool for understanding the nonlinear behavior of cancer progression and control. In this talk, we investigate a two-dimensional cancer growth model that has been enhanced by incorporating two biologically relevant mechanisms: tumor antigenicity and a strong Allee effect governing cancer cell survival thresholds. These additions allow for a more realistic representation of tumor establishment and immune system recognition. Our analysis begins with the identification and classification of the model’s equilibrium points across biologically feasible parameter ranges. We employ analytical and numerical techniques to determine stability and construct bifurcation diagrams. Particular attention is given to codimension-1 bifurcations, such as saddle–node and Hopf bifurcations, which signal qualitative transitions in tumor dynamics. We further extend the analysis to codimension-2 bifurcations, specifically Bogdanov–Takens, Bautin, and cusp bifurcations, to delineate regions in parameter space with complex dynamic behavior. We derive explicit conditions on key parameters that delimit dynamic regimes of tumor suppression, coexistence, and growth. Critical threshold curves are obtained that partition the phase space, revealing how the interplay between antigenicity and the Allee threshold influences tumor fate. The bifurcation structure identified in this enhanced cancer model provides a mechanistic framework for interpreting the three phases of cancer immunoediting, elimination, equilibrium, and escape in terms of underlying mathematical thresholds. These insights improve our theoretical understanding of tumor progression and may guide future therapeutic strategies that exploit immune response and tumor viability thresholds.

This work presents a computational methodology to determine the thermo‑kinetic parameters governing the fire behaviour of bio‑based thermal insulating materials, focusing on foams derived from flax and banana fibres. The methodology combines experimental mass‑loss‑rate (MLR) data from cone calorimeter tests at 50 kW/m² with a numerical inverse‑modelling framework implemented in the solid‑phase sub-model of Fire Dynamics Simulator (FDS). A simplified computational representation of the cone calorimeter is employed to accelerate simulations, enabling the use of the Shuffle Complex Evolution (SCE) optimization algorithm to identify the optimal set of parameters.

The proposed methodology estimates key thermal and kinetic parameters under a single‑reaction pyrolysis scheme. The optimized FDS models reproduce the experimental MLR curves with high accuracy, achieving mean‑squared errors of 1.36×10⁻⁴ kg·s⁻¹·m⁻² for flax foam and 1.02×10⁻⁴ kg·s⁻¹·m⁻² for banana foam. These results demonstrate the capability of the SCE‑driven inverse modelling framework to provide reliable material characterizations, obtaining appropriate thermos-kinetic parameters to reproduce decomposition reactions even in cases with limited prior information. Due to the optimization in the FDS modelling, and to the SCE approach, the proposed methodology requires only approximately one day of computation to identify optimal thermo-kinetic parameters governing decomposition reactions.

The methodology constitutes a robust applied‑mathematics tool for modelling fire dynamics in emerging sustainable insulation materials, offering a replicable workflow that supports material development, safety assessment, and integration into computational fire‑engineering simulations.

\begin{abstract}

\textbf{Introduction:}
Gradient-based optimization methods are widely used due to their simplicity and fast convergence. However, when applied to highly nonlinear and multimodal objective functions, these methods frequently converge to local optima because they rely exclusively on local gradient information. Chaotic optimization techniques provide an alternative mechanism for global exploration, yet they generally lack directional guidance and therefore may exhibit slow convergence during local refinement.

\textbf{Methods:}
To address these limitations, this paper proposes a novel Chaotic Gradient Method (CGM) that integrates gradient descent with density-controlled chaotic perturbations within a unified optimization framework. In contrast to existing hybrid chaotic approaches, where chaos is mainly used for initialization or stochastic exploration, the proposed method incorporates chaos directly into the gradient update rule. The key novelty lies in the introduction of nonlinear density transformation functions applied to chaotic sequences, which reshape their probability distribution and allow controlled perturbations around the gradient direction. This mechanism enables the algorithm to dynamically guide chaotic exploration toward promising regions of the search space while preserving the descent property of gradient-based optimization.

\textbf{Results:}
The performance of the proposed CGM is evaluated on several challenging multimodal benchmark functions, including Styblinski–Tang, Goldstein–Price, and Bukin functions. Numerical experiments demonstrate that the proposed approach improves the robustness of gradient-based optimization by increasing the success rate of reaching global optima and accelerating convergence compared with conventional gradient descent methods.

\textbf{Conclusions:}
The results confirm that density-controlled chaotic perturbations provide an effective mechanism for enhancing gradient-based optimization in complex multimodal landscapes. The proposed framework offers a new perspective for combining deterministic search with controlled chaotic dynamics. Future work will focus on adaptive selection of transformation functions and the application of the method to real-world optimization problems.

\end{abstract}

Hexagonal order recurs throughout nature as a locally energy-minimising arrangement for equal-radius interactions, from soap froths to bee combs to the densest circle packings. Logarithmic spirals, and in particular the golden-angle phyllotactic spiral, also arise across biological and physical systems, where new units accrete by a constant turn at multiplicatively increasing radii. This paper advances a coherent mathematical account linking these two motifs by proposing that the metric governing local interactions plays the decisive role. We formalise a hexagonal metric as a Minkowski norm whose unit ball is a regular hexagon and analyse constant-turn, multiplicative growth in this metric. We prove that such growth generates logarithmic spirals in the continuum limit and that the choice of the turn angle, maximising the asymptotic uniformity of placements, is uniquely achieved by the golden angle α* = 2π/φ², where φ = (1+√5)/2 is the golden ratio. The optimality criterion is expressed through minimal pairwise distance under the hexagonal norm, which we connect to the Diophantine properties of the turn angle. Specifically, we show that the golden ratio's extremal irrationality allows the spiral to best avoid the "resonant" directions of the sixfold anisotropic potential, thereby minimizing crowding. The theory explains why systems that experience local interactions with sixfold symmetry naturally express golden-angle spirals at mesoscopic scales, even when no overt hexagonal lattice is visible in Euclidean space. This yields testable predictions distinguishing this metric origin from purely Euclidean models, including specific parastichy counts, anisotropic Voronoi cell statistics, and the presence of spectral sidebands at multiples of π/3 in the angular structure factor.

A wildfire refers to an uncontrollable fire that spreads through vegetation, such as forests and grasslands, and poses a serious risk to ecosystems, infrastructure and human lives. Because wildfires are common throughout the world, it is very important to understand and model their spread so that we can better prepare our communities for their potential consequences. Most existing fire models, which were developed primarily, focus on only a few variables (i.e., wind velocity, slope, or Rothermel spread rate), making them less reliable in replicating how a wildfire propagates in reality. Therefore, this research presents a modified model for estimating the spatial extent of wildfires based on the use of the Level Set method and uncertainty. The factors considered in the modified model include wind speed and direction, slope, fuel type, moisture content, slope corrections, and uncertainty; and the slope corrector also considers slope steepness and tightness of the fuel pack in its definition, while the uncertainty considers the actual distance travelled from the ignition point. All computations were performed in MATLAB; and the shape of the fire was studied over time. Future studies will include detailed research on complicated ignition geometries, the effect of heterogeneity in the environment, and the impact of uncertainty on fire evolution, thus improving the dependability and utility of our model to predict and controlling wildfires. The study results demonstrate that the Level Set model can correctly depict the evolution of a fire's boundary.