Computer Science and Mathematics

Abstract: We construct a k-shadow complex Δ_k(N) from the nerve N of a convex sensor cover. We prove that the region covered by at least k sensors is homotopy equivalent to Δ_k(N), recovering the usual nerve-based coverage test when k=1 and giving a direct combinatorial method for redundant coverage. A synthetic network computation for k=1,2 agrees with direct geometric sampling, and the scripts and data are supplied for reproduction.

Abstract: Diffusion within porous media, such as biological tissues, often deviates from conventional Fick’s laws that may be described by space-fractional diffusion equations. Microscale tissue heterogeneity can be represented by a space-fractional Riesz Laplacian operator acting on the concentration. We consider a reaction-diffusion system with two spatial compartments – a proximal one of finite radius having a source, and an outer one extending to infinity where the source is absent but first-order decay takes place. The steady state is derived using Hankel and Mellin transforms, resulting in an integral kernels containing Bessel functions. We develop and compare three numerical quadrature methods for the Hankel transform: sinc quadrature, Ogata quadrature (based on Bessel zeros), and a hybrid asymptotic-numerical scheme. Numerical results and plots are presented for fractional exponents β=1/2,2/3,3/4 (2α=1+β). The integer-order case (α=1) is recovered as a limiting case. The hybrid method is about five times faster than the global quadratures for the same accuracy.

Abstract: Nonlinear systems of equations arise in a wide range of applications in engineering, physics, and biological modeling; however, classical Newton-type methods may fail when the initial approximation lies outside the basin of attraction of the desired solution. This work proposes a two-stage hybrid framework that couples Particle Swarm Optimization (PSO) for global exploration with the fifth-order Newton-Jarratt (NJN) iterative method for local refinement. The fifth-order convergence of the NJN phase, established through a complete Fréchet-derivative Taylor expansion with explicitly computed error constants, guarantees rapid local convergence once PSO delivers a sufficiently close starting point. The framework is validated on four test problems of increasing dimension (n=2,5,20,40): a two-dimensional benchmark algebraic system, a five-dimensional metabolic network model for ethanol production in Saccharomyces cerevisiae, and two large-scale systems arising from the discretization of a Hammerstein nonlinear integral equation. Over 30 independent runs per method, PSO-NJN achieves a 100% convergence rate in all four problems, with mean final residuals on the order 10⁻¹⁴-10⁻¹⁶. In comparison, pure PSO fails completely (0% success) on the high-dimensional Hammerstein cases (n=20,40) and achieves only 10% success on the metabolic model. These results confirm that combining global metaheuristic search with high-order local refinement yields a robust, scalable solver suitable for complex biological and engineering nonlinear systems.

Abstract: This paper presents a novel formulation of Vieta–Pell wavelets functions (VPW) constructed from classical Vieta–Pell polynomials through systematic scaling and translation techniques. The transformation from a global polynomial basis to a localized wavelet basis enables efficient representation of functions on the semi-open interval [0, 1), preserving orthogonality and compact support properties. The proposed wavelet system is employed to develop an operational matrix framework, which converts differential equations into algebraic systems. Based on this formulation, two numerical approaches are adopted for solving optimal control problems. The direct method is applied by transforming the performance index into a quadratic programming problem, which is solved using the Lagrange multiplier technique. This approach is utilized in engineering applications such as solar energy systems and power system protection, where optimal performance and stability are achieved. In addition, an indirect method based on Pontryagin’s Minimum Principle combined with spectral techniques is employed to solve a practical control problem related to dishwasher systems. The spectral formulation enhances the accuracy of the solution while maintaining computational efficiency. Theoretical properties of the proposed wavelets, including orthogonality, convergence, and accuracy, are rigorously established. Numerical results demonstrate that the Vieta–Pell wavelet approach provides high accuracy, flexibility, and efficiency in solving modern optimal control problems across various engineering domains.

Abstract: Claim reserving is one of the most important tasks in non-life insurance, as it directly affects solvency assessment, financial reporting, and risk management. Traditional reserving methods often assume a relatively homogeneous claim-development process and may fail to capture hidden structures within complex insurance portfolios. This paper introduces a novel reserving framework that integrates Topological Data Analysis (TDA) with both aggregate and micro-level reserving methodologies. Using a portfolio of motor insurance claim payments, we employ topological techniques to identify latent claim-development regimes and portfolio heterogeneity. The extracted topological information is subsequently incorporated into an Inverse Probability Weighting (IPW) reserving framework and a TDA-enhanced Chain-Ladder (CL) methodology. The empirical results suggest that the proposed TDA-based approaches may improve reserve estimation accuracy relative to their traditional counterparts. Both TDA-IPW and TDA-CL produce reserve estimates that are remarkably close to realized future claim payments. The findings suggest that topological structures contain valuable information for actuarial reserving and that Topological Data Analysis may provide a promising new direction for the development of reserving methodologies.

Abstract: A new method for finding the global extremum of functions of many variables under interval constraints is proposed. The method simulates the process of foraging for edible mushrooms (Boletus edulis, Leccinum aurantiacum, Leccinum scabrum, Cantharellus cibarius, Pleurotus, etc.) in a forest by a group of mushroom pickers. The algorithm includes a forest exploration stage (a set of feasible solutions) to find mushroom sites and an exploitation stage, during which previously identified mushroom sites are intensively explored by implementing various movement strategies for the mushroom pickers. The method is classified as both bioinspired metaheuristic algorithms and multi-agent evolutionary algorithms. Its effectiveness is demonstrated using eight typical problems of static parametric optimization of technical systems: pressure vessel problem, welded beam optimization problem, tension/compression spring design problem, gear train optimization design, speed reducer problem, three-bar truss problem, tubular column design optimization, flapping wing design optimization. Experience solving these problems allowed us to formulate recommendations for defining the hyperparameters of the proposed optimization method. The primary goal of the development was to apply this new bio-inspired optimization algorithm to three optimal control problems for discrete dynamic systems: optimal control of a single trajectory, a bundle of trajectories of a deterministic system starting from a given set of initial states, and optimal control of stochastic systems.

Abstract: In [1], Kulenovi'c, Ladas and Overdeep posed a conjecture asserting that every positive solution of the rational second-order difference equation \[ y_{n+1}=\frac{y_n(1+y_n)^2}{y_n(1+y_n)+(1+y_{n-1})},\qquad n=0,1,\ldots, \] converges to a finite limit. We confirm this conjecture by deriving a short identity showing that the sign of $y_{n+1}-y_n$ is invariant with respect to $n$, so every positive solution is monotone. A simple estimate then gives an explicit initial-data-dependent upper bound in the increasing case, while the decreasing case is bounded below by positivity. Hence every positive solution converges. In addition, we introduce the auxiliary sequence \[ t_n:=\frac{y_n(1+y_n)}{1+y_{n-1}}, \] which is monotone in the direction opposite to that of $y_n$. It yields nested two-sided enclosures of the limit and an exact invariant-series formula. Writing $g_n=t_n-y_n$ and $\rho_n=t_n/(1+t_n)^2$, we prove that \[ I_n=y_n+g_n\sum_{j=0}^{\infty}\frac{1}{1+t_{n+j}} \prod_{m=0}^{j-1}\rho_{n+m} \] is independent of $n$ and satisfies $I_n=L=\lim_{k\to\infty}y_k$. Hence the limiting equilibrium selected by the initial data is determined by the invariant value $I_0$.

Abstract: We study the third-order rational difference equation \[ z_{n+1}=\frac{\alpha+z_n}{Cz_{n-1}+z_{n-2}}, \qquad \alpha,C>0, \] with positive initial conditions. This equation is associated with a period-six trichotomy governed by the threshold $\alpha C^2=1$. We prove that the unique positive equilibrium is locally asymptotically stable exactly when $\alpha C^2>1$, is nonhyperbolic but not locally asymptotically stable at equality, and is unstable below the threshold. Our principal result resolves both the critical component of the trichotomy and the generalized period-six conjecture formulated by Amleh and Ladas: if $\alpha C^2=1$, then every positive solution converges, along its six residue classes, to a positive solution whose period divides six. Every such periodic limiting orbit is classified explicitly. In the normalized variables $x_n=Cz_n$, and after a cyclic choice of origin, it has the form \[ \left(a,ac,c,a^{-1},(ac)^{-1},c^{-1}\right),\qquad a,c>0, \] and every nonconstant member has prime period six. The proof combines the known boundedness theorem with an exact six-step factorization over $\mathbb Z[a,b,c,q]$. The source package separates certificate generation from a distinct certificate-reading verification stage, implemented in the same exact computer-algebra environment, and includes the expanded polynomials, hashes, software versions, and execution instructions. We also exclude positive prime period-two and period-three solutions in the full normalized parameter domain and derive a strict weighted reciprocal-mean entropy estimate for the unresolved supercritical problem.

Abstract: FEFLOW is used to analyze seepage flow through a tailings storage facility constructed by on-dam cycloning. Partial saturation of tailings beach material is accounted for by solving Richards’ transient flow equation throughout facility staged construction, using seepage analysis of idealized 1D and 2D staged construction processes to set FEFLOW time stepping and mesh size parameters. Computed results include design intended phreatic surface level, drain flows, and water balance of the tailings storage facility. Transient seepage analysis is also used to examine how as-built rise in the facility’s dam crest phreatic surface levels may be controlled by both hydraulic conductivity gradient of the tailings beaches and hydraulic conductivity of the dam downstream shells.

Abstract: This study investigates the unsteady magnetohydrodynamic (MHD) stagnation-point flow of a chemically reactive Rivlin-Ericksen viscoelastic nanofluid over a cylindrical surface with isothermal conditions and uniform nanoparticle concentration. The model incorporates induced magnetic field, thermal radiation, Joule heating, viscous dissipation, Brownian motion, thermophoresis, wall slip, and suction/injection effects. The governing nonlinear partial differential equations are transformed into coupled ordinary differential equations using similarity transformations. The novelty of this work lies in the simultaneous consideration of viscoelastic memory effects, nanoparticle transport with chemical reaction, induced magnetic field, and cylindrical stagnation-point geometry within a unified framework. A hybrid semi-analytical method combining the Homotopy Perturbation Method (HPM) and Galerkin weighted residual technique is employed to obtain approximate solutions that satisfy boundary conditions exactly. Results show that magnetic and viscoelastic parameters significantly reduce fluid velocity, while suction enhances boundary layer stability and transport rates. Thermal and concentration distributions are strongly influenced by radiation, viscous dissipation, Brownian motion, thermophoresis, and chemical reaction. The research advances understanding and applications of non-Newtonian transport over a cylindrical geometry and offers quantitative design guidance for applications in biomedical microdevices, polymer processing, and thermal management systems.

Abstract: This paper presents new families of triangular norms and conorms specifically constructed for type-2 fuzzy sets. Our approach considers several partially ordered sets of membership functions defined on the unit interval, each characterized by different structural properties and suited to representing particular kinds of uncertainty. For these settings, we define operators that consistently represent fuzzy intersection and union, filling gaps where no such type-2 operators were previously available. It is the first time that triangular norms and conorms are obtained with respect to both usual partial orders in the set of normal membership functions. Building on this framework, we integrate the proposed operators into a type-2 fuzzy time series. A large-scale evaluation on multiple benchmark time series shows that the new operators consistently achieve the best predictive accuracy, in terms of MAPE, in comparison with classical type-2 and type-1 fuzzy time series baselines. These results demonstrate that the algebraic design of type-2 operators has a direct impact on forecasting performance and can substantially improve the modeling of uncertainty in time-dependent data.

Abstract: This paper develops a new generalized model that describes the complex dynamical behavior of a financial system through three state variables, namely the interest rate, investment demand and the price index. The developed model extends and improves numerous financial models available in the literature by incorporating two time delays. The first delay accounts for the time lag in price adjustment, whereas the second captures the delayed feedback effect on investment demand. For the first time in the context of financial systems, a novel threshold parameter is introduced to characterize the existence of equilibria. The dynamical properties of the proposed model, including the stability and the occurrence of Hopf bifurcation, are rigorously analyzed. Furthermore, sensitivity analysis and numerical simulations are conducted to investigate the influence of model parameters on the dynamics of the financial system and to illustrate the analytical results.

Abstract: We develop a transfer-matrix framework for the asymptotic analysis of chromatic polynomials of recursively constructed prism-derived graph families. For each family {G_n}, the chromatic polynomial admits a finite spectral expansion P(G_n, z) = ∑_j α_j(z)λ_j(z)ⁿ, and the Beraha–Kahane–Weiss theorem identifies the limiting chromatic-root set through competition among dominant spectral branches. For the antiprism family A_n, the reduced transfer operator has discriminant ∆(z) = 9 − 4z. After the substitution w² = 9 − 4z, the unit-modulus conditions for the spectral branches λ_± pull back to the irreducible quartic curve x⁴ − 8x³ + 2x²y² + 21x² − 8xy² − 18x + y⁴ + 5y² = 0. The same quartic arises independently for the circulant family C_n({1, 2}) through an identical reduced spectral normal form. In contrast, the chord family C_n({1, 3}) satisfies λ+(z)λ−(z) = −1, forcing collapse of the reduced equimodular locus to the vertical segment ℜ(z) = 2, |ℑ(z)| ≤ 2. We further prove that the antiprism family possesses a nonfixed isolated accumulation point at the fifth Beraha number B₅ = (3 + √5)/2, arising from residual amplitude cancellation under unique dominant spectral behavior. The generalized Petersen family G(n, 2) exhibits an irreducible cubic spectral equation, demonstrating an obstruction to the quadratic pullback mechanism. These examples reveal a structural relationship between reduced transfer-matrix spectra and the algebraic geometry of limiting chromatic- root loci, showing that distinct recursive graph families may share identical asymptotic geometry through common reduced spectral normal forms.

Abstract: Exploratory factor analysis (EFA) and first-component scoring are widely used to rank samples in multi-indicator evaluation. A recurring empirical observation is that untreated negative indicators can make rankings appear unchanged when they are rare, unstable when positive and negative indicators are balanced, and reversed when negative indicators dominate. This paper gives a mathematical explanation of this phenomenon and separates two mechanisms that are often conflated. First, in a one-factor model, re-extracting the leading factor after a coordinate sign flip is invariant up to the unavoidable global sign of the factor; therefore, the population eigenstructure itself is not the source of the phase transition. Second, when scores are compared in a fixed semantic direction, or when EFA regression scores are published after an anchor-based sign-orientation rule, the common factor signal is partially cancelled by untreated negative indicators. We derive closed-form correlations, Kendall ordering probabilities, weighted thresholds for heterogeneous loadings, and a finite-sample critical band of order $n^{-1/2}$. Reproducible simulations verify the theory: a 200-repetition fixed-direction simulation shows a transition from correlation near $+1$ to near $0$ and then to near $-1$; EFA regression scores oriented by an anchor statistic switch direction at the predicted information-balance threshold; and a dedicated local-alternative experiment with $\Delta_I=c/\sqrt{n}$ confirms the predicted flip-probability scaling. The results justify mandatory same-direction preprocessing of negative indicators and recommend reporting weighted sign-balance diagnostics before ranking.

Abstract: In the context of high-density or ultra-low-temperature plasma environments,the mathematical characterization of equations for quantum plasmas is of fundamental necessity, as classical fluid models fail to capture the dominant quantum effects such as particle overlapping,wave packet interference and Pauli exclusion principle,which are critical for accurately describing the collective nonlinear behaviors of dust acoustic solitary waves in such extreme regimes.Previous studies on quantum dust acoustic waves have been extensive, with most relying on numerical simulations or perturbation expansions to predict wave profiles. Motivated by the sustained interest in this field,our work further explores the theoretical aspects of such solutions.The paper investigates the existence of positive concave solutions for a class of third-order nonlinear boundary value problems with Stieltjes integral nonlocal boundary conditions,which is derived from the quantum-corrected Korteweg-de Vries(KdV)equation governing the propagation of small-amplitude dust acoustic solitary waves in bounded quantum plasma systems.By constructing the corresponding Green's function and analyzing its structural properties, the monotone iterative technique is employed to establish the existence of positive solutions, and an explicit iterative scheme is further provided for the constructive computation of these solutions. A concrete numerical example is presented to validate the theoretical findings, which demonstrates that the constructed iterative sequences converge to the exact solution while preserving the positive and concave geometric properties of the wave profile.The obtained results not only enrich the qualitative theory of nonlocal differential equations, but also provide a solid theoretical foundation for the reliability of subsequent numerical investigations on quantum dust acoustic solitary waves,ensuring that the predicted stable wave profiles possess true physical reality.

Abstract: Plane tessellations naturally generate local combinatorial invariants such as vertex degree and vertex type, which encode essential geometric and topological information. In the present work, these invariants are used to construct complete bicolored graphs whose edge coloring is induced intrinsically by the equality or inequality of local tessellation properties. Two classes of structured Ramsey graphs are introduced: vertex-degree graphs and vertex-type graphs. Unlike classical Ramsey colorings, the proposed colorings are not arbitrary but arise from equivalence relations generated by local geometric invariants. Consequently, the resulting graphs possess a partition-generated or semi-transitive structure in which one color forms disjoint complete cliques corresponding to equivalence classes, while the complementary color connects distinct classes. Within this framework, a sharp Ramsey-type theorem is established: every such graph on five vertices necessarily contains a monochromatic triangle. The combinatorial core of the theorem reduces to the statement that among five values either three coincide or three are pairwise distinct. Turán numbers for the introduced structured colorings are calculated and shown to possess a characteristic asymmetry that is absent in classical extremal graph theory. The relation-ship between geometric symmetries of tessellations and automorphism groups of the associated Ramsey graphs is analyzed, demonstrating that tessellation symmetries induce graph automorphisms, while additional purely combinatorial symmetries may emerge from degree degeneracies. The results establish a direct connection between tessellation geometry, equivalence relations, Ramsey theory, and extremal graph theory.

Abstract: Actuarial valuation depends heavily on assumptions involving mortality, lapse behavior, and dis count rates. Small changes in these assumptions may produce material changes in liability values, leading to significant model risk for insurers and affecting long-term reserve adequacy and risk management decisions. This paper develops a functional-analytic framework for studying stability and sensitivity in de terministic actuarial valuation. The valuation process is formulated as a nonlinear operator acting on an infinite-dimensional space of model assumptions. Within this framework, we establish Lip schitz continuity, Fr´echet differentiability, and second-order sensitivity properties of the valuation operator under perturbations in mortality, lapse, and discount assumptions. Explicit first- and second-order sensitivity formulas are derived and interpreted in an actuarial setting. The theoretical results are illustrated through a numerical example using mortality data from the Social Security Administration actuarial life table. The numerical results show close agreement between exact perturbed valuations and the corresponding Fr´echet approximations. The framework developed in this paper provides a mathematically rigorous approach for studying valuation sensitivity, assumption uncertainty, and model risk in deterministic actuarial models.

Abstract: This article reviews the linear solvers available in OpenFOAM and assesses their impact on the convergence behaviour of the SIMPLE algorithm. The discretisation of transport equations in CFD results in large and sparse linear systems, for which the choice of linear solver strongly influences the computational time. Although the solver does not change the final discrete solution, the difference in speed and robustness between the solvers can be more than an order of magnitude. A brief overview is given of how the velocity and pressure fields are decoupled in OpenFOAM, followed by a detailed review of the main linear solver families, including direct methods, basic iterative methods, multigrid methods and Krylov subspace methods, with attention to their practical strengths and weaknesses. The performance of the most advanced solvers is evaluated on a full-scale non-reacting kiln case consisting of 2.3 million cells. The pressure-corrector equation is identified as the main bottleneck in the SIMPLE algorithm. The Conjugate Gradient (CG) solver with the Generalised Geometric–Algebraic MultiGrid (GAMG) preconditioner is found to be the fastest and most stable method, achieving speed-ups of up to a factor of 7 compared to the slower advanced methods. Using GAMG as preconditioner also improves the robustness of the Bi-CGStab method.

Abstract: This paper studies fixed-point iteration methods within the contraction-mapping framework and develops a center-based iterative algorithm constrained by Mahalanobis-distance rejection. Building on Banach’s fixed-point theorem, we relate contractivity to the Jacobian norm and derive estimators for the order of convergence, showing the method exhibits linear convergence under the stated conditions. The algorithm modifies standard k-means by using conditional expectations for parameter updates and discarding low-probability tail points of Gaussian components via a p-quantile criterion of the chi-squared Mahalanobis distribution. The procedure is analyzed in both continuous (marginal Gaussian) and discrete noisy settings, with numerical experiments that quantify convergence behavior and robustness to outliers. A spherical clustering validity index is introduced to optimize the p-quantile for pattern detection. Applications to industrial-scene, RGB segmented images demonstrate effective detection of spherical patterns in highly noisy structures, illustrating the method’s practical potential for robust Gaussian mixture estimation and pattern recognition.

Abstract: We rigorously disprove the existence of global smooth solutions to the three‑dimensional incompressible Navier–Stokes equations in a periodic rectangular cuboid domain $\Omega$ subject to the prescribed smooth body force. The smooth initial data of the flow field is derived from a two-dimensional stationary exact solution. The analysis is grounded in Sobolev space regularity, the decomposition of velocity into a time‑averaged mean flow and a disturbance flow, the local vanishing of sum of the viscous term and the body force, and the Energy–Velocity Monotonicity Principle (EVMP). When the Reynolds number exceeds the critical value for turbulent transition, the nonlinear convective term dominates over the viscous term and external force term, nonlinear interactions amplify disturbances, leading to local cancellation of the sum of the mean flow viscous term and the body force with the disturbance viscous term at a finite critical time $t^*>0$ and interior point $\boldsymbol{x}^*\in\Omega$. This cancellation leads to the local mechanical energy gradient along the streamline being zero when the time derivativer is zero, which by EVMP requires $|\boldsymbol{u}(\boldsymbol{x}^*,t^*)|=0$, contradicting the existing non‑vanishing velocity. The contradiction generates a finite‑time regularity singularity, under which the velocity gradient $L^\infty$‑norm diverges. This violates the Sobolev embedding condition required for global smoothness of solutions. This study resolves the problem statement (D) in Fefferman (2006).