This paper investigates the concept of locally harmonious coloring in the context of high-dimensional interconnection net works, specifically the hypercube Qn, its structural variants such as the folded hypercube FQn, the augmented cube AQn, and the crossed cube CQn. We aim to determine χlh(Qn), χlh(FQn), χlh(AQn), and χlh(CQn), and to analyze how dimensional variations and structural augmentations influence their locally harmonious colorability. Furthermore, we establish relationshipsbetween χlh and other graph invariants such as degree, diameter, and automorphism group symmetry. The study provides new insights into the combinatorial structure of hypercube-based networks and their applications in parallel architectures, fault tolerant communication, and distributed computation.
First and foremost, securing the electoral process is fundamental to public trust in democratic governance. Traditional voting systems, from paper ballots to electronic machines, have long survived despite vulnerabilities to tampering, limited auditability, and an absence of strong end-to-end verifiability. To address these, this paper proposes the Secure and Transparent Blockchain Voting Algorithm (STBVA), a blockchain-based voting system powered by Proof of Authority (PoA) consensus, elliptic curve cryptography (ECC), and homomorphic encryption. Accordingly, PoA can support deterministic fast block production with low computational overhead, which makes it suitable for regulated election environments. In particular, ECC offers efficient authentication for users, while Paillier homomorphic encryption keeps votes private during tallying processes without revealing individual ballots. The proposed system is implemented and evaluated on a permissioned blockchain network consisting of seven validators and 20 full nodes, accommodating up to 500,000 simulated voters. Experimental results show that confirmation latency is 1.2 s at median, the sustained throughput is 1200 tps, and the accuracy of end-to-end vote recording is 99.8\%. Thereafter, a formal attacker model, security claims, and proof sketches corroborate the resilience against forgery, double voting, and ledger manipulations. The results underpin that STBVA is able to achieve scalable, privacy-preserving, and tamper-resistant election infrastructure to cater for national-level online voting.
Two-person games have been extensively studied in classical game theory, but uncertainty and imprecision in real-world scenarios necessitate more advanced mathematical frameworks. Pentapartitioned Neutrosophic sets, provide a powerful tool for handling contradiction, ignorance, unknown, and inconsistent information. This paper explores two-person games in a Pentapartitioned Neutrosophic environment, where players’ strategies, payoffs, and outcomes are expressed using Pentapartitioned Neutrosophic numbers. We present fundamental definitions, solution concepts, and equilibrium conditions tailored for such games. The findings demonstrate that Pentapartitioned Neutrosophic game theory provides a more flexible and realistic approach to strategic interactions involving indeterminacy.
This paper introduces neutrosophic nano β-continuous functions and explores their fundamental properties in neutrosophic nano topological spaces. The study establishes their role in extending continuity concepts for handling uncertainty and indeterminacy.
This paper investigates boundary value problems for the fractional Pauli operator on a finite square domain, addressing a significant gap in the literature where such problems have not been previously studied. The fractional Pauli operator generalizes the standard Pauli operator by replacing the classical Laplacian with the fractional Laplacian (−∆)α=2, introducing non-local quantum effects. We employthe spectral definition of the fractional Laplacian on bounded domains, expanding the solution as a double trigonometric series that automatically satisfies Dirichlet boundary conditions. The problem is reduced to solving a linear algebraic system for the series coefficients, for which we prove existence and uniqueness in appropriate fractional Sobolev spaces. Numerical experiments for various fractional ordersα demonstrate significant deviations from the classical case (α = 2), with solutions exhibiting enhanced amplitudes and diffusive characteristics as α decreases. Rigorous convergence analysis establishes the continuous transition to the classical Pauli operator as α ! 2−.
In this paper, we study the mean ergodic theorems and the weighted ones on locally compact hypergroups. Among other obtained results, for the class of all commutative hypergroups $\mathcal{H}$ with a Plancherel measure $\widetilde{\omega}$ that ${\rm supp}(\widetilde{\omega})=\widehat{\mathcal{H}}$, we prove that if $\left(k_{j}\right)_{j \in \mathbb{N}}$ is a subsequence of $\mathbb{N}$, $f\in L^2(\mathcal{H})$, and $\mu$ is a power bounded measure on $\mathcal{H}$ such that the sequence$$\left(\frac{1}{m} \sum_{n=1}^{m}\underbrace{\mu\ast\ldots\ast\mu}_{k_{n}-\text{times}}\ast f\right)_{m\in\mathbb{N}}$$weakly converges in $L^2(\mathcal{H})$, then the numerical sequence $\left(\frac{1}{m} \sum_{n=1}^{m} \alpha^{k_n}\right)_{m\in\mathbb{N}}$ is convergent too for all $\alpha\in \mathbb{C}$ with $\tilde{\omega}\left(\{\xi\in\widehat{\mathcal{H}}:\hat{\mu}(\xi)=\alpha\}\right)>0$.
This study examines an economic model and explores the Hopf bifurcation by individually varying the indebtedness factor and the output-capital ratio parameter. Both analytical and numerical methods are used to determine the conditions and coefficients for the normal form of the Hopf bifurcation. The critical coefficient for this bifurcation is identified using the central manifold theory. Additionally, the phase portrait of the model near the critical values of the indebtedness factor and the output-capital ratio parameter is illustrated using the Matcont software package.
The thermal diffusion coefficient in radioactive materials is not a constant value, and this makes the heat transfer problem different in such materials and materials that are not homogeneous and have been destroyed or are disintegrating in some way. In the problem under discussion, the heat diffusion coefficient is time-dependent and satisfies a nonlinear integral equation. The existence and uniqueness of the solution to the integral equation in question are discussed in detail in Chapter 13 of the Book "Encyclopedia of the One-Dimensional Heat Equation" by Cannon, J. R. The integral equation in question is not a standard Volterra integral equation and therefore has not been studied much from a numerical perspective. For example, if we apply the fixed point method, which is a powerful tool in the analysis of existence and uniqueness, discussed in Chapter 13 of the aforementioned Book, to a numerical solution, we cannot go even one step forward with this method. Since the unknown function is located at the kernel of a nested integral, applying canonical methods becomes difficult. Therefore, in this paper, we have discussed a hybrid method of numerical integration and iterative methods that solves the problem with sufficient accuracy. In Section 5, we have extracted several sample problems using the properties of the heat equation in the case where the thermal diffusivity is Time-dependent. The numerical solution of these sample problems in the Section 6 demonstrates the efficiency and accuracy of the proposed method.
This paper addresses Multi-Criteria Group Decision Making (MCGDM), also known as Multiple Attribute Group Decision Making (MAGDM), under the framework of intuitionistic fuzzy sets. To solve fuzzy linear algebraic equations, linear space techniques involving real eigenvalues are employed. These solutions are then used to determine decision-maker weights in MAGDM problems. During the weight determination process, multiple criteria are explicitly incorporated, and several results obtained through the proposed methods are normalized. Additionally, decision-maker weights for attributes, along with corresponding decision-making approaches, are introduced. Furthermore, Artificial Neural Network (ANN) techniques are applied to enhance the determination of decision-maker weights. The feasibility and effectiveness of the proposed approach are demonstrated through numerical examples. The convergence curve shows stable error reduction without underfitting or overfitting, validating the robustness of the proposed ANN framework for reliable application in intuitionistic fuzzy set–based MAGDM.
This work introduces a novel set, the Intuitionistic Complex Fuzzy Set (ICFS), that expands traditional intuitionistic fuzzy sets into a complex-valued domain and captures interacting attributes more effectively in the decision support framework based on ICFS. A new aggregation operator called the Intuitionistic Complex Fuzzy Einstein Correlated Geometric (ICFECG) operator and a new score and accuracy function for the ICFS are proposed and to ensure theoretical robustness, rigorous proofs are provided for multiple theorems associated with the newly developed ICFECG operator, the score and the accuracy functions. This operator effectively combines expert opinions while preserving both the amplitude and phase components of complex uncertainty, thereby ensuring that the aggregated information accurately reflects the full structure of the intuitionistic complex fuzzy evaluations. To improve efficiency in solving MAGDM problems, a data mining–based dimensionality reduction strategy that helps identify and remove redundant or weakly influential attributes is introduced. Artificial Neural Network (ANN) techniques are also incorporated to enhance the learning ability and optimization of the decision-support process. A new defuzzification function is proposed to integrate all the ICFS components, yielding a crisp value for enhancing the data mining and ANN computations. The final hybrid model combines ICFS theory, the ICFECG operator, data mining, and ANN optimization which effectively handles high-dimensional, correlated, and uncertain information arising in the decision making environment. A numerical case study shows that our methodology reduces the computational load, removes insignificant alternatives, and significantly improves decision accuracy, stability, and reliability.
The main purpose of this paper is to present a novel concept of separation axioms in neutrosophic topological spaces by means of neutrosophic N_tr Λ_P-open sets. The concepts of neutrosophic N_tr Λ_P-T_i spaces(i=0,1,2) are introduced and their properties are studied.
The utilization of the Indian rail system has grown at a very high rate and there is one of the largest train track networks in the world in the country. Despite the creation of various sophisticated means of transport, congestion, inefficiency and bad connectivity remain factors to contend with. To overcome these challenges, the metro rail has beendiscovered to be the most possible urban mass transit system and can be easily modeled using graph theory with vertices represented by stations and edges by tracks. In this paper, we begin by examining traditional metrics like connectivity, complexity, diameter,average distance between the terminals and potential expansion of the network in the hopeof quantifying passenger convenience and efficiency. We also advance the research withnew concepts: vertex and edge domination are used to compute the minimum criticalstation for effective surveillance, vertex and edge connectivity to quantify survivability against failure and labeling or coloring techniques for use with scheduling, traffic control and resource allocation. This joint approach results in both classical and new findings for more resilient metro network planning and construction.
Benzenoid systems are formed by collections of congruent hexagons arranged in the plane such that any two hexagons are either disjoint or share a common edge. These structures are naturally studied through graph-theoretic packing parameters. For a fixed graph H, an H-packing of a graph G is a family of vertex-disjoint subgraphs of G, each is isomorphic to H. In this work, we determine the P3- packing number and an induced P3-packing k-partition number for three standard benzenoid families: the triangular benzenoid system, the rhombic benzenoid system, and the zigzag benzenoid system. For each class, algorithmsare provided for computing these parameters, together with justification of their correctness. The results yield exact values for the corresponding packing and partition numbers in these benzenoid structures.
The Birkhoff polytope graph can be considered as the Cayley graph of the symmetric group $S_n$ with respect to $\mathcal{C}_n$, the set of cycles in $S_n$. Since the degree of every Cayley graph is a natural bound on several parameters of the graph, in this note by presenting a formula for $|\mathcal{C}_n|$, the degree of the Birkhoff polytope graph, we prove that it is bounded from above by$\lfloor e\big{(}(n-1)! +(n-2)!+(n-3)! +\cdots 1 \big{)}\rfloor$, where $e$ is the Neper number.
This paper introduces a novel operational matrix approach for addressing a class of fractional-order optimal control problems, where the derivative is taken in the regularized Prabhakar sense. The method employs Bernoulli polynomials and utilizes their operational matrix of regularized Prabhakar derivative and inherent properties to convert the original problem into a finite-dimensional optimization problem. Using the Lagrange multiplier approach, the required optimality conditions are derived, yielding an algebraic system from the original problem. Solving this system yields an approximate fractional optimal solution. The practicality and efficiency of the proposed approach are confirmed via a series of numerical examples.
The present study examines hypercube graphs through the lens of Čech rough closure theory (ČRCT). Leveraging the binary representation and inherent symmetry of hypercubes, we introduce a closure-based approach for modelling uncertainty and vagueness in discrete spaces. Our method constructs set approximations from vertex adjacency, offering a systematic way to examine neighbourhood configurations. To illustrate the applicability of this framework, we present examples on low-dimensional hypercubes. The findings highlight how rough closure concepts can enrich graph-theoretic analysis and provide flexible tools for developing models in discrete mathematics.
A more flexible and nuanced view of space and objects that are inherently ambiguous or inaccurate is made possible by soft nano topology. Connectivity concepts play an major role in topological spaces. The classification and comprehension of the various topological spacestructures is aided by connectivity. The main aspect of this article is to introduce new types of soft nano connected space known as soft nano ic-pre generalized connected space anddiscuss its properties. Further, we introduce new type of hyper connected space known as soft nano ic-pre generalized hyper connected space and look over its characteristics. This idea makes it instinctive to comprehend topological structures and some difficult theorems, and it aids in the development of concepts like compactness and separation axioms in soft nano topology.
Let G = (V, E) be a connected graph and γvb(G) denotes the valency based domination number of G or simply vb-domination number of G. In this paper, analogous to isolated vertex in domination, defined valency based isolated vertex (or simply, vb-isolated vertex ) in vb-domination and proved that vb-isolated vertices belong to any vb-dominating set. Also studied its properties in the graph G as well as in its Mycielskian, µ(G). Established an inequality connecting vb-domination number of Mycielskian of G and domination number of G. Obatined an upper bound for vb-domination number of Mycielskian of regulargraphs. Also calculated vb-domination number of Mycielskian of some graph classes.
This research explores a closure Filter structure which generates a novel closure Filter topology and defines a new operator that satisfies Kuratowski's closure axioms. Also, we investigate their relationships with generalized topological spaces by defining a local closure function and discussing their basic properties and characterizations.
Unmanned Aerial Vehicles (UAVs) are used rapidly in different fields. Few of the important areas where UAVs are essential in disaster management and agriculture owing to their cost-effectiveness and accessibility to remote areas. However, adverse weather conditions like rain hinder their navigation. This study is a deep learning approach using MobileNetV2 to detect rainy conditions from UAV captured images. It aims to enhance the operational safety and efficiency. The balanced dataset of 245Kimages across seven rain classes was used. The dataset was divided into training, testing, and validation sets in the ratio 70:15:15. This convolutional neural network model achieved a test accuracy of 95.35%. This suggests that the model is reliable and robust and can be further researched for real-time deployment.
