Search Results (104)

This thesis pioneers the development of q-Rung Neutrosophic Fuzzy Rough Sets (q-RNFRSs), establishing the first theoretical framework that integrates q-Rung Neutrosophic Sets with rough approximations to break through the conventional ${\mu }^q+{\eta }^q+{\nu }^q\le 1$ constraint of existing fuzzy–rough hybrids, achieving unprecedented capability in extreme uncertainty representation through our generalized model (${\mathcal{T}}^q+{\mathcal{I}}^q+{\mathcal{F}}^q\le 3$). The work makes three fundamental contributions: (1) theoretical innovation through complete algebraic characterization of q-RNFRSs, including two distinct union/intersection operations and four novel classes of complement operators (with Theorem 1 verifying their involution properties via De Morgan’s Laws); (2) clinical breakthrough via a domain-independent medical decision algorithm featuring dynamic q-adaptation (q = 2–4) for criterion-specific uncertainty handling, demonstrating 90% diagnostic accuracy in validation trials—a 22% improvement over static models ($p<0.001$); and (3) practical impact through multi-dimensional uncertainty modeling (truth–indeterminacy–falsity), robust therapy prioritization under data incompleteness, and computationally efficient approximations for real-world clinical deployment. Full article

In this study, we focus on solving the nonlinear time-fractional Hirota–Satsuma coupled Korteweg–de Vries (KdV) and modified Korteweg–de Vries (MKdV) equations, using the Yang transform iterative method (YTIM). This method combines the Yang transform with a new iterative scheme to construct reliable and efficient solutions. Readers can understand the procedures clearly, since the implementation of Yang transform directly transforms fractional derivative sections into algebraic terms in the given problems. The new iterative scheme is applied to generate series solutions for the provided problems. The fractional derivatives are considered in the Caputo sense. To validate the proposed approach, two numerical examples are analysed and compared with exact solutions, as well as with the results obtained from the fractional reduced differential transform method (FRDTM) and the q-homotopy analysis transform method (q-HATM). The comparisons, presented through both tables and graphical illustrations, confirm the enhanced accuracy and reliability of the proposed method. Moreover, the effect of varying the fractional order is explored, demonstrating convergence of the solution as the order approaches an integer value. Importantly, the time-fractional Hirota–Satsuma coupled KdV and modified Korteweg–de Vries (MKdV) equations investigated in this work are not only of theoretical and computational interest but also possess significant implications for achieving global sustainability goals. Specifically, these equations contribute to the Sustainable Development Goal (SDG) “Life Below Water” by offering advanced modelling capabilities for understanding wave propagation and ocean dynamics, thus supporting marine ecosystem research and management. It is also relevant to SDG “Climate Action” as it aids in the simulation of environmental phenomena crucial to climate change analysis and mitigation. Additionally, the development and application of innovative mathematical modelling techniques align with “Industry, Innovation, and Infrastructure” promoting advanced computational tools for use in ocean engineering, environmental monitoring, and other infrastructure-related domains. Therefore, the proposed method not only advances mathematical and numerical analysis but also fosters interdisciplinary contributions toward sustainable development. Full article

Number sequences are among the research areas of interest in both number theory and linear algebra. In particular, the study of matrix representations of recursive sequences is important in revealing the structural properties of these sequences. In this study, the relationships between the elements of the k-Fibonacci and k-Oresme sequences were analyzed using matrix algebra through matrix structures created by connecting the characteristic equations and roots of these sequences. In this context, using the properties of these matrices, the identities $A_n^2-A_{n+1}{A}_{n-1}=k^{-2n}$, $A_n^2-A_n{A}_{n-1}+{\displaystyle \frac{1}{k^2}}{A}_{n-1}^2=k^{-2n}$, and $B_n^2-B_n{B}_{n-1}+{\displaystyle \frac{1}{k^2}}{B}_{n-1}^2=-(k^2-4)k^{-2n}$, and some generalizations such as $B_{n+m}^2-(k^2-4)A_{n-t}{B}_{n+m}{A}_{t+m}-(k^2-4)k^{2t-2n}{A}_{t+m}^2=k^{-2m-2t}{B}_{n-t}^2$, $A_{t+m}^2-B_{t-n}{A}_{n+m}{A}_{t+m}+k^{2n-2t}{A}_{n+m}^2=k^{-2n-2m}{A}_{t-n}^2$, and more were derived, where $m,n,t\in \mathbb{Z}$ and $t\ne n$. In addition to this, the solution pairs of the algebraic equations $x^2-B_{p}xy+k^{-2p}{y}^2=k^{-2q}{A}_p^2$, $x^2-(k^2-4)A_{p}xy-(k^2-4)k^{-2p}{y}^2=k^{-2q}{B}_p^2$, and $x^2-B_{p}xy+k^{-2p}{y}^2=-(k^2-4)k^{-2q}{A}_p^2$ are presented, where $A_p$ and $B_p$ are k-Oresme and k-Oresme–Lucas numbers, respectively. Full article

In this paper, we investigate the module algebra structures of $X_q\left(A_1\right)$ on the quantum polynomial algebra ${\mathbb{C}}_q[x,y,z]$. The automorphism group of ${\mathbb{C}}_q[x,y,z]$ is denoted by $\mathrm{Aut}\left({\mathbb{C}}_q[x,y,z]\right)$, and we give a complete classification of $X_q\left(A_1\right)$-module algebra structures on ${\mathbb{C}}_q[x,y,z]$, where $K_1,K_2\in \mathrm{Aut}\left({\mathbb{C}}_q[x,y,z]\right)$. Full article

This article introduces the theory of three-variable q-truncated exponential Gould–Hopper-based Appell polynomials by employing a generating function approach that incorporates q-calculus functions. This study further explores these polynomials by using a computational algebraic approach. The determinant form, recurrences, and differential equations are proven. Relationships with the monomiality principle are given. Finally, graphical representations are presented to illustrate the behavior and potential applications of the three-variable q-truncated exponential Gould–Hopper-based Appell polynomials. Full article

This paper extends the previous work by the author on m-null pairs of operators in Hilbert space. If an elementary operator L has elementary symbols A and B that are p-null and q-null, respectively, then L is $(p+q-1)$-null. Here, we prove the converse under strictness conditions, modulo some nonzero multiplicative constant—if L is strictly $(p+q-1)$-null, then a scalar $\lambda \ne 0$ exists such that $\lambda A$ is strictly p-null and ${\lambda }^{-1}B$ is strictly q-null. Our constructive argument relies essentially on algebraic and combinatorial methods. Thus, the result obtained by Gu on m-isometries is recovered without resorting to spectral analysis. For several operator classes that generalize m-isometries and are subsumed by m-null operators, the result is new. Full article

This paper explores the common neighborhood energy ($E_{CN}\left(\mathsf{\Gamma }\right)$) of graphs derived from the dihedral group $D_{2n}$ and generalized quaternion group $Q_{4n}$, specifically the non-commuting graph (NCM-graph) and the commuting graph (CM-graph). Studying graphs associated with groups offers a powerful approach to translating algebraic properties into combinatorial structures, enabling the application of graph-theoretic tools to understand group behavior. The common neighborhood energy, defined as the sum of the absolute values of the eigenvalues of the common neighborhood (CN) matrix, i.e., ${\sum }_{i=1}^p\left|{\zeta }_i\right|$, where ${\left\{{\zeta }_i\right\}}_{i=1}^p$ are the CN eigenvalues, provides insights into the structural properties of these graphs. We derive explicit formulas for the CN characteristic polynomials and corresponding CN eigenvalues for both the NCM-graph and CM-graph as functions of n. Consequently, we establish closed-form expressions for the $E_{CN}$ of these graphs, which are parameterized by n. The validity of our theoretical results is confirmed through computational examples. This study contributes to the spectral analysis of algebraic graphs, demonstrating a direct connection between the group-theoretic structure of $D_{2n}$ and $Q_{4n}$, as well as the combinatorial energy of their associated graphs, thus furthering the understanding of group properties through spectral graph theory. Full article

This paper presents a unique multi-Q valued bipolar picture fuzzy set (MQVBPFS) methodology to tackle issues in cybersecurity risk assessment under conditions of ambiguity and contradicting data. The MQVBPFS framework enhances classical fuzzy theory through three key innovations: (1) multi-granular Q-valued membership, (2) integrated bipolarity for representing conflicting evidence, and (3) refined algebraic operations, encompassing union, intersection, and complement. Contemporary fuzzy set methodologies, such as intuitionistic and image fuzzy sets, inadequately encapsulate positive, negative, and neutral membership degrees while maintaining bipolar information. Conversely, our MQVBPFS architecture effectively resolves this restriction. Utilizing this framework for threat assessment and risk ranking, we create a tailored cybersecurity algorithm that exhibits 91.7% accuracy (in contrast to 78.2–83.5% for baseline methods) and attains 94.6% contradiction tolerance in empirical evaluations, alongside an 18% decrease in false negatives relative to conventional approaches. This study offers theoretical progress in fuzzy set algebra and practical enhancements in security analytics, improving the handling of ambiguous and conflicting threat data while facilitating new research avenues in uncertainty-aware cybersecurity systems. Full article

Semiring partitioning is widely used in mathematics, computer science, and data analysis. The purpose of this paper is to add to the theory of semirings by proposing a novel method to construct Q-ideals for partitioning Boolean semirings. We introduce the set of all seeds—all s-tuples over a particular Boolean algebra—and the notion of their weight and complement. Utilizing this new method for constructing Q-ideals, we develop a nested hierarchical partitioning algorithm based on the weight of selected seeds. Additionally, we determine the maximal semiring homomorphism corresponding to this proposed method. Full article

The concept of strong atoms in Q-algebras is discussed herein. In this work, some properties of strong atoms are provided. We show that there is no Q-algebra X with $\left|X\right|\ge 3$ such that all elements are strong atoms. We also find that any two-element subset of X containing a constant 0 is a subalgebra of X whenever X contains a strong atom. Moreover, any subset of X with the cardinality equal to 3 containing a strong atom and a constant 0 is always a subalgebra. We present some results concerning the concept of an ideal. In a Q-algebra X that contains a strong atom, any ideal of X is a subalgebra of X. An ideal of a Q-algebra X that is induced by any subset containing a strong atom is equal to X. Furthermore, we show that, for any Q-algebra X with a strong atom, there is only one ideal containing a strong atom. In particular, for $\left|X\right|\le 4$, we propose that a finite union of ideals of X is again an ideal of X. Full article

To systematically address the intricate multiple criteria decision-making (MCDM) challenges to practical situations where uncertain and hesitant information plays a critical role in guiding optimal choices. In this article, we introduce the concept of $\mathfrak{m}$-polar Q-hesitant fuzzy ($\mathcal{MPQHF}$) $\mathfrak{B}\mathrm{C}\mathbb{K}/\mathfrak{B}\mathrm{C}\mathbb{I}$ algebras, combining $\mathfrak{m}$-PFS theory with Q-hesitant fuzzy set theory in the framework of $\mathfrak{B}\mathrm{C}\mathbb{K}$/$\mathfrak{B}\mathrm{C}\mathbb{I}$ algebras. This innovative approach enhances the attitudes of uncertainty, vagueness, and hesitance of data in decision-making processes. We investigate the features and actions of this proposed hybrid approach to fuzzy sets and hesitant fuzzy sets, focusing on $\mathcal{MPQHF}$ subalgebras, and explore the characteristics of several kinds of ideals under $\mathfrak{B}\mathrm{C}\mathbb{K}$/$\mathfrak{B}\mathrm{C}\mathbb{I}$ algebras. It also showed that it can better represent complex levels of uncertainty than regular sets. The proposed method’s theoretical framework offers a better way to show uncertain data in areas like engineering, computer science, and computational mathematics. By linking theoretical advancements of $\mathcal{MPQHF}$ sets with practical applications, we highlight the benefits and challenges of this approach. Demonstrating the practical uses of the $\mathcal{MPQHF}$ sets aims to encourage broader adoption. Symmetry plays a vital role in algebraic structure and is used in various fields like decision-making, encryption, pattern recognition problems, and automata theory. Furthermore, this work enhances the understanding of algebraic structures and offers a robust tool for career exploration and development through improved decision-making methodologies. Full article

This paper deals with several nonlinear partial differential equations (PDEs) of mathematical physics such as the concatenation model (perturbed concatenation model) from nonlinear fiber optics, the plane hydrodynamic jet theory, the Kadomtsev–Petviashvili PDE from hydrodynamic (soliton theory) and others. For the equation of nonlinear optics, we look for solutions of the form amplitude Q multiplied by $e^{i\Phi }$, $\Phi$ being linear. Then, Q is expressed as a quadratic polynomial of some elliptic function. Such types of solutions exist if some nonlinear algebraic system possesses a nontrivial solution. In the other five cases, the solution is a traveling wave. It satisfies Abel-type ODE of the second kind, the first order ODE of the elliptic functions (the Weierstrass or Jacobi functions), the Airy equation, the Emden–Fawler equation, etc. At the end of the paper a short survey on the Jacobi elliptic and Weierstrass functions is included. Full article

We explore nonlocal symmetries in a class of Hamiltonian dynamical systems governed by second-order differential equations. Specifically, we establish an algorithm for deriving nonlocal symmetries by utilizing the Jacobi metric and the Eisenhart–Duval lift to geometrize the dynamical systems. The geometrized systems often exhibit additional local symmetries compared to the original systems, some of which correspond to nonlocal symmetries for the original formulation. This novel approach allows us to determine nonlocal symmetries in a systematic way. Within this geometric framework, we demonstrate that the second-order differential equation $\ddot{q}-F\left(q\right)=0$ admits an infinite number of nonlocal symmetries generated by the infinite-dimensional conformal algebra of a two-dimensional Riemannian manifold. Applications to higher-dimensional systems are also discussed. Full article

An inverse-square probability mass function (PMF) is at the Newcomb–Benford law (NBL)’s root and ultimately at the origin of positional notation and conformality. $Pr\left(Z\right)={\left(2Z\right)}^{-2}$, where $Z\in {\mathbb{Z}}^{+}$. Under its tail, we find information as harmonic likelihood $\mathcal{L}\left(\left[s,t\right)\right)=H_{t-1}-H_{s-1}$, where $H_n$ is the nth harmonic number. The global $\mathbb{Q}$-NBL is $Pr\left(b,q\right)={\displaystyle \raisebox{1ex}{$\mathcal{L}\left(\left[q,q+1\right)\right)$}\!\left/ \!\raisebox{-1ex}{$\mathcal{L}\left(\left[1,b\right)\right)$}\right.}={\left(q{H}_{b-1}\right)}^{-1}$, where b is the base and q is a quantum ($1\le q<b$). Under its tail, we find information as logarithmic likelihood $\ell \left(\left[i,j\right)\right)=ln{\displaystyle \raisebox{1ex}{$j$}\!\left/ \!\raisebox{-1ex}{$i$}\right.}$. The fiducial $\mathbb{R}$-NBL is $Pr\left(r,d\right)={\displaystyle \raisebox{1ex}{$\ell \left(\left[d,d+1\right)\right)$}\!\left/ \!\raisebox{-1ex}{$\ell \left(\left[1,r\right)\right)$}\right.}={log}_r\left(1+{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$d$}\right.}\right)$, where $r\ll b$ is the radix of a local complex system. The global Bayesian rule multiplies the correlation between two numbers, s and t, by a likelihood ratio that is the NBL probability of bucket $\left[s,t\right)$ relative to b’s support. To encode the odds of quantum j against i locally, we multiply the prior odds ${\displaystyle \raisebox{1ex}{$Pr\left(b,j\right)$}\!\left/ \!\raisebox{-1ex}{$Pr\left(b,i\right)$}\right.}$ by a likelihood ratio, which is the NBL probability of bin $\left[i,j\right)$ relative to r’s support; the local Bayesian coding rule is $\tilde{o}\left(j:i|r\right)={\displaystyle \raisebox{1ex}{$i$}\!\left/ \!\raisebox{-1ex}{$j$}\right.}{log}_r{\displaystyle \raisebox{1ex}{$j$}\!\left/ \!\raisebox{-1ex}{$i$}\right.}$. The Bayesian rule to recode local data is $\tilde{o}\left(j:i|r^{\prime }\right)=\tilde{o}\left(j:i|r\right){\displaystyle \raisebox{1ex}{$lnr$}\!\left/ \!\raisebox{-1ex}{$ln{r}^{\prime }$}\right.}$. Global and local Bayesian data are elements of the algebraic field of “gap ratios”, ${\displaystyle \frac{A-B}{C-D}}$. The cross-ratio, the central tool in conformal geometry, is a subclass of gap ratio. A one-dimensional coding source reflects the global Bayesian data of the harmonic external world, the annulus $\left\{x\in \mathbb{Q}|1\le \left|x\right|<b\right\}$, into the local Bayesian data of its logarithmic coding space, the ball $\left\{x\in \mathbb{Q}|\left|x\right|<1-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$b$}\right.}\right\}$. The source’s conformal encoding function is $y={log}_r\left(2x-1\right)$, where x is the observed Euclidean distance to an object’s position. The conformal decoding function is $x={\displaystyle \frac{1}{2}}\left(1+r^y\right)$. Both functions, unique under basic requirements, enable information- and granularity-invariant recursion to model the multiscale reality. Full article

The aim of this work is to demonstrate that all linear derivatives of the tensor algebra over a smooth manifold M can be viewed as specific cases of a broader concept—the operation of derivation. This approach reveals the universal role of differentiation, which simplifies and generalizes the study of tensor derivatives, making it a powerful tool in Differential Geometry and related fields. To perform this, the generic derivative is introduced, which is defined in terms of the quantities $Q_k^{\left(i\right)}\left(X\right)$. Subsequently, the transformation law of these quantities is determined by the requirement that the generic derivative of a tensor is a tensor. The quantities $Q_k^{\left(i\right)}\left(X\right)$ and their transformation law define a specific geometric object on M, and consequently, a geometric structure on M. Using the generic derivative, one defines the tensor fields of torsion and curvature and computes them for all linear derivatives in terms of the quantities $Q_k^{\left(i\right)}\left(X\right)$. The general model is applied to the cases of Lie derivative, covariant derivative, and Fermi derivative. It is shown that the Lie derivative has non-zero torsion and zero curvature due to the Jacobi identity. For the covariant derivative, the standard results follow without any further calculations. Concerning the Fermi derivative, this is defined in a new way, i.e., as a higher-order derivative defined in terms of two derivatives: a given derivative and the Lie derivative. Being linear derivative, it has torsion and curvature tensor. These fields are computed in a general affine space from the corresponding general expressions of the generic derivative. Applications of the above considerations are discussed in a number of cases. Concerning the Lie derivative, it is been shown that the Poisson bracket is in fact a Lie derivative. Concerning the Fermi derivative, two applications are considered: (a) the explicit computation of the Fermi derivative in a general affine space and (b) the consideration of Freedman–Robertson–Walker spacetime endowed with a scalar torsion field, which satisfies the Cosmological Principle and the computation of Fermi derivative of the spatial directions defining a spatial frame along the cosmological fluid of comoving observers. It is found that torsion, even in this highly symmetric case, induces a kinematic rotation of the space axes, questioning the interpretation of torsion as a spin. Finally it is shown that the Lie derivative of the dynamical equations of an autonomous conservative dynamical system is equivalent to the standard Lie symmetry method. Full article