Abstract: The Collatz conjecture, despite its deceptively simple formulation, remains one of the most enduring unsolved problems in mathematics. It posits that repeatedly applying the operation — divide by 2 if even, or multiply by 3 and add 1 if odd — to any positive integer will eventually reach the number 1. While the conjecture has been numerically validated for vast ranges, a general proof has eluded mathematicians.This paper introduces a novel structural and visual framework for understanding the Collatz problem by constructing a "Collatz Tree" — a directed rooted tree that systematically organizes all natural numbers. Each branch originates from an odd number and extends through its powers of two, forming infinite geometric sequences. We rigorously prove that every natural number is uniquely contained within this tree structure.Furthermore, we demonstrate that constructing a tree via the reverse Collatz operation (starting from 1 and applying valid inverses of the Collatz function) reproduces the exact same structure as the Collatz Tree. This equivalence implies that any number, when followed downward through the Collatz process, ultimately converges to the root node 1.By reframing the conjecture through this structural lens, we reveal a new avenue for understanding the convergence behavior of Collatz sequences, providing clarity to the flow of natural numbers through a deterministic tree topology, and reinforcing the conjecture’s validity through structural completeness and absence of cycles.

Abstract: The convergence order of Jarratt-type methods for solving nonlinear equations are obtained without using the Taylor expansion. We use assumptions on the derivatives of the involved operator up to second order only contrary to the earlier studies. The proof provided in this paper does not depend on the Taylor series expansion which in turn reduces assumptions on the higher order derivatives of the involved operator and increases the applicability of these methods. The applicability of the method is further extended using the concept of generalized condition in the local convergence and majorizing sequences in the semi-local analysis. Numerical examples and Basins of attractions of the methods are provided in this study.

Abstract: We introduce a novel mathematical framework that unifies classical smoothness theory with fractal regularity, addressing a long-standing gap between Sobolev spaces and multifractal structures. This hybrid function space is shown to be complete and to enjoy optimal embedding properties into classical functional settings. A central contribution lies in the local geometric characterization of functions via scale-sensitive dimension metrics, enabling explicit delineation between smooth and fractal behavior. The theory is further strengthened by spectral stability results for operator approximations and by perturbation-resilient estimations of local irregularity. Numerical experiments validate the framework across diverse contexts, including texture segmentation in image analysis, robustness in spectral discretizations, and the modeling of multifractal signals. This work offers a unified analytical and computational toolbox for heterogeneous systems exhibiting abrupt transitions in regularity across space or time.

Abstract: We introduce the notion of noncommutative equiangular lines and derive noncommutative versions of fundamental van Lint-Seidel relative and Gerzon universal bounds.

Abstract: This study presents an innovative solution method for ultra-fine group slowing-down equations tailored to stochastic media with double heterogeneity (DH), focusing on advanced nuclear fuels such as Fully Ceramic Microencapsulated (FCM) fuel and Mixed Oxide (MOX) fuel. Addressing the limitations of conventional resonance calculation methods in handling DH effects, the proposed UFGSP method integrates the Sanchez-Pomraning technique with ultra-fine group transport theory to resolve spatially dependent resonance cross-sections in both matrix and particle phases. The method employs high-fidelity geometric modeling, iterative cross-section homogenization, and flux reconstruction to capture neutron self-shielding effects in randomly distributed media. Validation across seven FCM fuel cases, four poison particle configurations (BISO/QUADRISO), and four plutonium spot problems demonstrates exceptional accuracy, with maximum deviations in effective multiplication factor keff and resonance cross-sections remaining within ±138 pcm and ±2.4%, respectively. Key innovations include the ability to resolve radial flux distributions within TRISO particles and address resonance interference in MOX fuel matrices. The results confirm that UFGSP significantly enhances computational precision for DH problems, offering a robust tool for next-generation reactor design and safety analysis.

Abstract: Traditional portfolio optimization techniques predominantly rely on the classical mean–variance framework introduced by Markowitz, which focuses on balancing expected returns against risk, typically measured by variance. However, in volatile and structur-ally unstable markets such as cryptocurrencies, this approach often fails to capture the full spectrum of uncertainty and diversification potential. This paper introduces an al-ternative methodology grounded in entropy, a fundamental concept in information theory that quantifies uncertainty and disorder. By incorporating entropy into the portfolio optimization process, we offer a more generalizable, distribution-free approach that enhances diversification and resilience.We develop and analyze three distinct en-tropy-based models: the maximum Shannon entropy model, the second-order entropy (Tsallis) model, and the maximum weighted Shannon entropy model. These formula-tions extend the traditional mean–variance approach by integrating nonlinear uncer-tainty measures, enabling a richer representation of investor preferences and asset in-terdependencies. Analytical solutions to the proposed models are derived using the method of Lagrange multipliers, ensuring mathematical rigor and interpretability.The proposed models are empirically validated using a portfolio composed of four leading cryptocurrencies—Bitcoin (BTC), Ethereum (ETH), Solana (SOL), and Binance Coin (BNB)—with market data from January to March 2025. The case studies demonstrate how entropy-based optimization leads to well-diversified portfolios, robust under market turbulence and heavy-tailed return distributions. Notably, the models facilitate dynamic adjustments in asset allocation in response to shifts in return–risk characteristics and entropy levels. This study contributes to the ongoing generalization of portfolio theory by positioning entropy as both a diversification enhancer and a structural risk measure. It provides theoretical insight, practical tools for asset allocation in high-volatility environments, and paves the way for future research in entropy-driven financial optimization frameworks.

Abstract: Maximum Rank Distance (MRD) codes are optimal error-correcting codes that attain the Singleton-like bound for rank distance. Despite significant progress in constructing MRD codes over finite fields, a general theory remains elusive. This paper explores a novel approach to constructing MRD codes using Particle Swarm Optimization (PSO), a population-based metaheuristic algorithm well-suited to complex, high-dimensional optimization problems. By leveraging PSO’s search capabilities, we propose an algorithmic framework for discovering MRD codes with novel parameters. This approach aims to fill gaps in the existing literature by providing a systematic method for identifying MRD codes where algebraic constructions are not yet known.

Abstract: This paper presents a systematic comparative study of three major axiomatic set theory systems: Zermelo-Fraenkel system with “the Axiom of Choice” (ZFC), von Neumann-Bernays-Gödel system (NBG), and Morse-Kelley system (MK). The research begins by tracing the historical development and motivations behind these three axiomatic frameworks, followed by a detailed analysis of their axiom structures and fundamental concepts. The systems are then compared across several dimensions: expressive power, metamathematical properties, and practical applications. The innovative aspect of this study lies in introducing the Coq proof assistant as a formal verification tool to implement and compare MK, NBG, and ZFC, systematically investigating their differences within a formalized environment. Research findings indicate that the three axiomatic systems present distinct advantages and challenges during formalization, providing new perspectives for understanding the essential characteristics of axiomatic set theory and its position in mathematical foundations. This comparative analysis helps clarify the relationships between these three axiomatic systems and provides a theoretical reference for future formal verification work in set theory.

Abstract: This study focused on the transformation of an exponentially growing divergent function sin(Rln(x)) into a convergent function by its complementary exponential function xt in such a manner that the sizes of positive and negative areas under sin would be the same. The transformation will provide the entire sin function with self-compensatory behavior. The exponent's value was compute and found that it equals to -1/2, which is the only exponent, which lets entire product of the function converge to zero (sum of area for positive real numbers and sum of products for natural numbers). The exponent -1/2 is algebraically and geometrically inevitable for the function xtsin(Rln(x)) converging to zero. The result directly affects the critical line position in the Euler-Riemann zeta function.

Abstract: The article constructs all anisotropic spaces of four-dimensional numbers in which commutative and associative operations of addition and multiplication are defined. In this case, so-called "zero divisors" appear in these spaces. The structures of zero divisors in each space are described and their properties are investigated. It is shown that there are two types of zero divisors and they form a two-dimensional subspace of the four-dimensional space. A space of 4x4 matrices is constructed that is isomorphic to the space of four-dimensional numbers. The concept of the spectrum of a four-dimensional number is introduced and a bijective mapping between four-dimensional numbers and their spectra is constructed. Thanks to this, methods for solving linear and quadratic equations in four-dimensional spaces are developed. It is proved that a quadratic equation in a four-dimensional space generally has four roots. The concept of the spectral norm is introduced in the space of four-dimensional numbers and the equivalence of the spectral norm to the Euclidean norm is proved.

Abstract: Given the extensive application of anti-Hermitian matrices in engineering and the sciences, this paper presents the general solution to a constrained system of matrix equations that incorporate their anti-Hermitian property. The primary focus of this study is to solve this constrained system of quaternion matrix equations. Solvability conditions are established using rank equalities, and explicit representation formulas are provided, employing the Moore-Penrose inverse of coefficient matrices and their projections. An algorithm and a numerical example are presented to validate the research findings. The numerical example utilizes a unique direct method for obtaining solutions to the given system, based on exclusive determinantal representations of the Moore-Penrose inverse within the noncommutative row-column determinant theory recently developed by one co-author. The results obtained demonstrate significant novelty, even when applied to the corresponding complex matrix equations as a special case.

Abstract: As the flipped classroom model gains traction in mathematics education, questions remain about its effectiveness across diverse classroom settings. This study surveyed 266 senior/upper secondary second-year students to gain insights into their experiences of learning mathematics in flipped classrooms during an academic term. It assessed their perspectives on the model’s implementation, strategies utilized, opportunities and challenges encountered, and suggestions for improvement. The research obtained data through questionnaires, classroom observations, and semi-structured interviews. Quantitative data from the questionnaires underwent processing with descriptive statistics and the Wilcoxon Signed-Rank test for paired samples, while qualitative data from observations and interviews passed through thematic analysis. The findings demonstrate a statistically significant improvement in participants’ experiences post-intervention, highlighting the positive impact of flipped classrooms on student engagement and academic achievement. Although students report challenges related to time management and self-paced learning, they value the autonomy and flexibility provided by video lessons and pre-class learning resources. The study suggests that, while the flipped classroom model holds promise for improving student engagement and achievement in senior secondary mathematics, it is paramount to address issues like planning, resource availability, technology access, teacher preparation, and student participation for success. The study further recommends that teachers provide structured guidance for pre-class tasks and establish supportive classroom environments to help students adapt to this technology-mediated model.

Abstract: This paper presents a method for constructing polynomial Lyapunov functions to analyze the stability of nonlinear dynamical systems. The approach is based on Genetic Programming, a variant of Genetic Algorithms where the search space is consists hierarchical tree structures. In our formulation, these polynomial functions are represented as binary trees. The Lyapunov conditions for exponential stability are interpreted as a minimax optimization problem, using a carefully designed fitness metric to ensure positivity and dissipation within a chosen domain. The Genetic Algorithm then evolves candidate polynomial trees, minimizing constraint violations and continuously refining stability guarantees. Numerical examples illustrate that this methodology can effectively identifies and optimizes Lyapunov functions for a wide range of systems, indicating a promising direction for automated stability proofs in engineering applications.

Abstract: The application of digital inclusive finance in various industries, particularly in rural areas, is gaining significant attention. The traditional agricultural sector, which focuses on rural labor economics, is more sensitive to financial innovations due to geographical and other constraints. This paper investigates how digital inclusive finance affects rural labor economics by integrating the Improved Gravitational Search Algorithm Random Forest (IGSA-RF) with the Gini coefficient, Out-of-Bag (OOB) coefficient, and the Gini-OOB coupling coefficient. For empirical analysis, Jiangsu Province, China, is selected as the research subject. The findings suggest that: (1) Digital inclusive finance has a long-term positive impact on consumption, gross regional product, and the average wage index of rural workers; (2) There is a growing trend in agricultural machinery power over time. However, the study found that gender, age, and the development of labor-intensive industries did not show significant improvement. To further enhance rural labor economics, the study builds a framework based on the comprehensive application of digital finance, focusing on those in rural areas who are vulnerable to fluctuations in financial markets.

Abstract: This paper proposes an exact (O(n^2)) polynomial-time method of solving the Travelling Salesman Problem (TSP) and building the shortest possible route iteratively. First, building the route for a small number of cities closest to the cities’ centroid or their location’s geometric center and then expanding the route further to more distanced cities until all the cities are joined to the route. This method introduces a measure called Excess Path (EP) that is used to determine how and where a new city must be joined to already existing shortest possible route so the new expanded route may also be considered the shortest. This paper also compares the number of computations needed using this method to the number of computations needed using Brute-force search method and shows this method’s big advantage over the Brute-force search.

Abstract: The paper advances the new technique for constructing the exceptional differential polynomial systems (X-DPSs) and their infinite and finite orthogonal subsets. First, using Wronskians of Jacobi polynomials (JPWs) with a common pair of the `indexes, we generate the Darboux-Crum nets of the rational canonical Sturm-Liouville equations (RCSLEs). It is shown that each RCSLE in question has four infinite sequences of quasi-rational solutions (q-RSs) such that the polynomial components from each sequence form a X-Jacobi DPS composed of simple pseudo-Wronskian polynomials (p-WPs). For each p-th order rational Darboux Crum transform of the Jacobi-reference (JRef) CSLE used as the starting point, we formulate two rational Sturm-Liouville problems (RSLPs) by imposing the Dirichlet boundary conditions on the solutions of the so-called ‘prime’ SLE at the ends of the intervals (-1,+1) or (+1,∞). Finally, we demonstrate that the polynomial components of the q-RSs representing the eigenfunctions of these two problems have the form of simple p-WPs composed of p Romanovski-Jacobi (R-Jacobi) polynomials with the same pair of the indexes and a single classical Jacobi polynomial or accordingly p classical Jacobi polynomials with the same pair of positive indexes and a single R-Jacobi polynomial. The common fundamentally important feature of all the simple p-WPs involved is that they do not vanish at the finite singular endpoints –the main reason of why they were selected for the current analysis in the first place. The discussion is accompanied by a sketch of the one-dimensional quantum-mechanical problems exactly solvable by the aforementioned infinite and finite EOP sequences.

Abstract: This work focuses on investigating rough Marcinkiewicz integrals associated to specific surfaces. Whenever the kernel functions belong to Lq(Sm−1) space, the Lp boundedness of these Marcinkiewicz integrals is confirmed. This finding along with Yano’s extrapolation argument prove the Lp boundedness of the aforementioned integrals under weaker conditions on the kernels. The results in this work improve and generalize various previously known results on Marcinkiewicz integrals.

Abstract: This article introduces a new fuzzy double integral transformation called the fuzzy double Yang transformation. We review some of the main properties of the transformation and find the conditions for its existence. We prove the theorems for partial derivatives and fuzzy unitary convolution. All new results are applied to find an exact solution to the fuzzy parabolic Volterra integro-differential equation with a memory kernel. In addition, a numerical example is provided to illustrate the accuracy and superiority of the proposed method with the help of symmetric triangular fuzzy numbers.

Abstract: This paper advances the foundations of tensor and category theories by introducing novel concepts and rigorous constructive proofs. We generalize tensor theory through the innovative notion of a generalized tensor index, a versatile framework that unifies diverse tensor indices, and explore its transformation properties. Using fractional derivatives, we provide a geometrical interpretation of these generalized tensors, revealing new insights into their structure. Additionally, we forge a deep connection between tensor and category theories, integrating sets, tensors, categories, and functors with extensions like partial differentiation and integration. This synthesis yields original constructs—setorial tensors, categorial tensors, and functorial tensors—which open uncharted pathways in mathematical analysis. Our contributions not only extend prior research but also significantly enhance tensor theory, category theory, set theory, logic, topology, algebraic geometry, foundations, and philosophy, with potential applications spanning physics, geometry, and beyond.

Abstract: Recently the exponential arithmetic-geometric index ($EAG$) was introduced. The exponential arithmetic-geometric index ($EAG$) of a graph $G$ is defined as $$EAG(G)=\sum_{v_iv_j \in E(G)}\,e^\frac{d_i + d_j}{2 \sqrt{d_i d_j}},$$ where $d_i$ represents the degree of the vertex $v_i$ in $G$. The characterization of extreme structures in relation to graph invariants from the class of unicyclic graphs is an important problem in discrete mathematics. Cruz, Rada and Sanchez [Extremal unicyclic graphs with respect to vertex-degree-based topological indices, {\it MATCH Commun. Math. Comput. Chem.\/} {\bf 88} (2022) 481--503] proposed a unified method for finding extremal unicyclic graphs for exponential degree-based graph invariants. However, in the case of $EAG$, this method is insufficient to generate the maximal unicyclic graph. Consequently, the same article presented an open problem for the investigation of the maximal unicyclic graph with respect to this invariant. This article completely characterizes the maximal unicyclic graph in relation to $EAG$.