Neutrosophic statistics encompasses several mathematically distinct frameworks that are frequently conflated, blurring its relation to interval and classical methods. We organize neutrosophic statistics as three layers: (i) neutrosophic arithmetic 𝑁 = 𝑎 + 𝑏𝐼 with indeterminate components beyond intervals; (ii) neutrosophic set statistics over (𝑇, 𝐼, 𝐹) triplets as structurally independent dimensions; and (iii) plithogenic statistics with degree of appurtenance. We provide six lines of validation: a formal analysis showing interval projection is a lossy compression of neutrosophic numbers; empirical evidence on 510 expertannotated triplets that (𝑇, 𝐼, 𝐹) data does not reduce to an interval-derived construction (T-I coupling 3.4× weaker than the algebraic counter-factual, I-F sign reversal, and 34.9% paraconsistency 𝑇 + 𝐹 > 1 which interval-derived data cannot reach); a head-to-head benchmark on ten causal hypotheses where neutrosophic zone classification reaches 90% accuracy against expert ground truth; a UCI benchmark on five datasets with Consensuszone accuracy 0.973–1.000; an extended evaluation across fourteen UCI datasets and five base classifiers with Friedman 𝜒2 = 44.55, 𝑝 < 10−8; and layer-selection diagnostics. The thirdanswer library provides an open-source implementation. The framework supplies principled tools for multi-criteria decision analysis and uncertainty-aware evaluation.
During the production process, defective production rates as well as inspection processes become essential, as customers have become increasingly attentive to the quality of the products they receive. This paper aims to analyze a queueing-inventory system in which both the production and inspection processes are imperfect. The novelty of this study lies in the combination of these two imperfections in a single inventory-queueing system. The inspection process involves two kinds of misclassification errors: Type-I occurs when a non-defective item is identified as defective, while Type-II occurs when a defective item is incorrectly labeled as non-defective. Two storage facilities of limited capacity are considered, one allocated for all produced items and the other reserved for items deemed non-defective after inspection, and a production process controlled by an (s, S)-policy. The proposed model is formulated as a continuous-time Markov chain and analyzed by the Matrix Geometric Method. The stability condition is derived, and several key performance measures are obtained. The cost function is then constructed and minimized with respect to production, inspection, and service rates using the Particle Swarm Optimization algorithm. The numerical study shows that the probability of defective production has a considerable impact on all performance measures and that the capacity of the second depot affects both the mean number of customers in the system and the total cost. The results indicate that production, inspection, and service rates must be adjusted jointly according to the probability of defect and the capacity of the depot to achieve the minimum cost.
Fractional calculus has recently emerged as a powerful mathematical framework for modelling complex systems. Despite its mathematically intricate foundation, it serves as a generalization of classical differentiation andintegration, extending these operations to arbitrary, non-integer (fractional) orders. This study investigates the identification of time-dependent coefficient in a time-fractional Bi-flux equation using interior point measurement within the spatial domain, employing the contraction mapping principle and explore numerical solutions by creating numerical algorithms and testing examples.
Given a subset $A \subseteq \mathbb{R}$, the Hattori topology $H(A)$ is a hybrid of the Euclidean and lower-limit topologies, giving every point $a \in A$ a base of Euclidean neighborhoods $(a-\ve, a+\ve)$ and every point $b \not\in A$ a base of lower-limit neighborhoods $[b, b+\ve)$. Here we consider hybrid topologies from the lower-limit and right-ray topologies on $\mathbb{R}$ as well as hybrid topologies formed from at least one Alexandroff topology on $\mathbb{R}$.
This study builds on the research by Collins et al. [Topology Appl. 154 (2007) 2312-2322], Aytar [Numer. Funct. Anal. Optim. {29} (3-4) (2008) 291-303] and Kostyrko et al. [Real Anal. Exchange {26} (2) (2000/2001) 669-686] to introduce the concepts of forward (resp. backward) rough I-convergence, as well as forward (resp. backward) rough I-Cauchy sequences, in asymmetric metric spaces. In this context, we would like to emphasize the following points:[(i):] A forward (resp. backward) rough I-convergent sequence may not necessarily be a forward (resp. backward) rough I-Cauchy sequence.[(ii):] We address an open problem proposed by Das et al. [Page-561, Math. Slovaca, 63 (3), 2013, 545-562].Simultaneously, we introduce forward (resp. backward) rough I-limit points, as well as forward (resp. backward) rough I-cluster points, within the context of asymmetric metric spaces to characterize an ideal and the metrizability of asymmetric metric spaces. Furthermore, we demonstrate the non-existence of any sequence $x=\{x_n\}_{n\in\mathbb{N}}$ for which a forward (resp. backward) closed set $F$ does not coincide with the forward (resp. backward) rough I-cluster point set of that sequence.
The concept of perfect $\mathcal{E}$-error-correcting codes constitutes a fundamental benchmark in classical coding theory; however, a corresponding definition for these codes remains absent in the quantum domain. This study addresses this theoretical gap by formally defining perfect $\mathcal{E}$-error-correcting quantum codes and constructing a new family of codes that satisfy this definition. We derive these quantum codes from classical linear codes generated over residue class rings of prime Hurwitz integers. The newly constructed quantum codes are presented with their parameters, and those qualifying as perfect $\mathcal{E}$-error-correcting codes are identified. Additionally, this study designs quantum logic gates tailored for the proposed codes over Hurwitz integers. These findings extend the algebraic utility of Hurwitz integers in quantum information theory and provide a novel class of quantum codes equipped with their fundamental logical operators.
Motivated by the development of Bernstein--Kantorovich operators that preserve affine functions, we propose a new class of Baskakov--Kantorovich type operators $\mathcal{V^*}_{n,m}$, along with the associated subfamilies $\mathcal{V}^*_{n,m}$, $\mathcal{V}^*_{n,2m-1}$, and $\mathcal{V}^*_{n,2m}$. The approximation behavior of these operators is thoroughly investigated and compared with that of classical Baskakov--Kantorovich operators $V_{n}$. Our results demonstrate that the proposed operators, Particularly $\mathcal{V}^*_{n,2m-1}$ exhibit superior approximation properties while preserving affine functions. Moreover, we establish that for the operators $\mathcal{V}^*_{n,m}$, the moments satisfy\[\mathcal{V}^*_{n,m}(t^i,x) \to V_n(t^i,x) \quad \text{as } m \to \infty, \quad i=1,2,\ldots\]Several Voronovskaja-type theorems are also derived for the newly introduced operators. In addition, we show that the operators $\mathcal{V}^*_{n,m}$ possess important shape-preserving properties. Finally, numerical examples and graphical illustrations are presented to validate and visualize the theoretical findings.
Let $G$ be a finite group. A subgroup $H$ of $G$ is called weakly $S \Phi$-supplemented in $G$ if $G$ has a subgroup $T$ such that $G=H T$ and $H \cap T \leq \Phi(H) H_{s G}$, where $H_{sG}$ is the subgroup generated by all those subgroups of $H$ that are $s$-permutable in $G$. We investigate the structure of finite groups in which certain minimal subgroups satisfy the weakly $S \Phi$-supplemented property, obtainsome new characterizations of nilpotent groups and improve several recent theorems.
This study dedicated to provide numerical solutions of fractionalRayleigh-Stokes problem in two-dimension (2D). The problem has Riemann-Liouvilletype fractional derivative. To obtain time discretization for theconsidered problem, we use a difference scheme along with an integrationprocedure. Hence a time semi-discretized scheme is obtained. We examinestability and convergence of the time semi-discretized scheme. Thenwe developed a local radial basis function partition of unity methodfor discretization of space variables. By combining time and spacediscretization we acquire a full discrete scheme for solving the consideredproblem. We solve some test problems and compare obtained resultsfor the test problems with existing techniques in the literature,to show efficiency of the proposed method.
In this paper, we investigate the well-posedness and the asymptotic behavior of solutions to piezoelectric beams under magnetic, delay, and thermodiffusion effects. First, we establish the existence and uniqueness of solutions by applying semigroup theory. Then, using a perturbed energy method and constructing suitable Lyapunov functionals, we prove the exponential stability of the system.
Cartan numbers form a four-dimensional real algebra with a mixedhyperbolic--nilpotent structure and an indefinite quadratic character, whichprecludes the induced modulus from defining a norm and distinguishes theirmatrix theory from that of classical hypercomplex systems.In this paper, we develop a systematic framework for matrices over the algebraof Cartan numbers by introducing a complex $2\times 2$ representation thatestablishes an algebra isomorphism between $\mathbb{S}$ and a Hermitian-typesubalgebra of $M_{2}(\mathbb{C})$.Based on this representation, we define the complex adjoint matrix$\Xi_{\mathbf{K}}$, prove the identity$\Xi_{\mathbf{K}\mathbf{M}}=\Xi_{\mathbf{K}}\Xi_{\mathbf{M}}$, and show that theinvertibility of $\mathbf{K}$ is equivalent to that of $\Xi_{\mathbf{K}}$,leading to the definition of a $q$-determinant with multiplicativity andsimilarity invariance properties.We further investigate right eigenvalues and generalized right eigenvalues ofCartan number matrices and matrix pencils, demonstrating that non-real Cartaneigenvalues generate infinite similarity orbits and that the complex spectraof $\Xi_{\mathbf{K}}$ fully characterize eigenvalues in $\mathbb{C}$.
We will undertake a detailed analysis of the set-theoretical complexity of general relative multifractal measures, together with the corresponding dimension functions. This investigation clarifies the set-theoretic complexity of these objects and provides a rigorous foundation for studying their behaviour in a broad multifractal framework.
In recent works, the concept of quotient proximal groups was introduced. This paper introduces the concept of strongly $\mathbb{R}$-factorizable proximal groups and employs the theory of quotient proximal groups to prove an important theorem which states that every compact proximal group is strongly $\mathbb{R}$-factorizable. We also define $\omega$-narrow proximal groups and prove that every strongly $\mathbb{R}$-factorizable proximal group is $\omega$-narrow.
In this paper, we study theorder of the Dirichlet and Neumann eigenvalues for theSturm-Liouville equation $$-(p(x)y')'+q(x)y=\lambda\rho(x) y.$$ We prove that $\lambda_1\geq\mu_2$ (resp. $\lambda _{1}\leq\mu_2 $) for single-barrier $q$ and single-well $p\rho$ in$[0,1]$ (resp. for single-well $q$ and single-barrier $p\rho$ in $[0,1]$).
In recent years, fractional–order derivatives and integrals have proven tobe powerful tools for modeling complex systems with memory effects, mak-ing them well–suited for describing various real–world phenomena. Fractional–order models are often more effective than their integer–order counterpartsin capturing the dynamics of materials such as polymers, biological systems,and other physical processes. Given these considerations, the study of qual-itative properties, particularly Ulam–type stabilities, of fractional integro–differential equations has gained significant attention in mathematical research due to their wide–ranging applications in aerodynamics, biology, physics, electrodynamics, polymer science, and other disciplines. Motivated by this,we investigate various types of Ulam–type stabilities, including Ulam–Hyersstability, Ulam–Hyers–Rassias stability, and σ–semi–Ulam–Hyers stability,for mild solutions of noninstantaneous impulsive fractional–order integro–differential equations with local conditions in a Banach space. By introduc-ing a novel set of suitable assumptions, we establish stability results andobtain five new findings concerning these stability concepts. Our analysisconsiders both finite and infinite intervals. As an application, we presentthree illustrative examples that support the theoretical findings. The anal-ysis in this work relies on the Banach fixed–point theorem and the Bieleckimetric, which are fundamental tools in nonlinear analysis with applicationsin functional equations, fractal theory, and optimization. Our findings con-tribute to the ongoing development of stability theories for fractional–ordermodels and their applications in various scientific fields.
In many studies based on Bigeometric Calculus, an approximation to the BigeometricTaylor series is used without knowing the correct version. The reason for that can be seen easily inthe proof of the Bigeometric Taylor Series presented in the current paper. Based on this Taylor seriesthe Bigeometric Runge-Kutta method is derived explicitly. As the Runge-Kutta method is based onthe first derivative, it is not surprising that the coefficients and parameters in the Bigeometric RungeKutta behave according to the Butcher Tableau. The convergence and stability tests are also appliedto the Bigeometric Runge-Kutta. Application of the Bigeometric Runge-Kutta method to problemswith known closed form solutions show the superiority of this method for a certain family of problemscompared to the one in Newtonian calculus. Furthermore, the Bigeometric Runge-Kutta method isapplied to mathematical modelling in biology and the Bigeometric R¨ossler attractor, showing thegeneral applicability of the method
Let $\sigma=\{{\sigma_i|i\in I}\}$ be some partition of the set of all primes $\mathbb{P}$ and $G$ a finite group. A group is said to be \textit{$\sigma$-primary} if it is a finite $\sigma_i$-group for some $i$.A subgroup $H$ of $G$ is said to be: \textit{$\sigma$-subnormal} in $G$ if there exists a subgroup chain $H=H_0\leq H_1\leq \cdots \leq H_n=G$ such that either $H_{i-1}$ is normal in $H_i$ or $H_i/(H_{i-1})_{H_i}$ is $\sigma$-primary for all $i=1,\ldots,n$; \textit{$\sigma$-permutably embedded} in $G$ if $H$ is $\sigma$-full and for every $\sigma_i\in \sigma(H)$, every Hall$\sigma_i$-subgroup of $H$ is also a Hall $\sigma_i$-subgroup of some $\sigma$-permutable subgroup of G.We say that a subgroup $H$ of $G$ is: \textit{$m$-$\sigma$-permutably embedded} in $G$ if$H=\langle A, B\rangle$ for some modular subgroup $A$ and $\sigma$-permutably embedded subgroup $B$ of $G$;\textit{weakly $m$-$\sigma$-permutably embedded} in $G$ if there are an $m$-$\sigma$-permutably embedded subgroup $L$ and a$\sigma$-subnormal subgroup $T$ of $G$ such that $G=HT$ and $H\cap T\leq L \leq H$.
We introduce generalized Lipschitz classes $\mbox{Lip}_k(\beta)$ and $\mbox{lip}_k(\beta)$ of functions associated with the Sturm-Liouville operator $L$; and we prove two versions of Boas-type theorems for the Sturm-Liouville transform $\mathscr{F}$. An application to Sturm-Liouville multipliers is given.
Given a graph $\mathcal{G}$ on $\mathfrak{n}$ vertices and diameter $d$, a radio $k$-coloring $\mathfrak{h}$ is an assignment of non-negative integers to the vertices of $\mathcal{G}$ such that for every pair of vertices $\mathfrak{u}$ and $\mathfrak{v}$, the condition $|\mathfrak{h}(\mathfrak{u}) - \mathfrak{h}(\mathfrak{v})| \geq 1 + k - \mathfrak{d}(\mathfrak{u}, \mathfrak{v})$ holds, where $\mathfrak{d}(\mathfrak{u}, \mathfrak{v})$ is the distance between vertices $\mathfrak{u}$ and $\mathfrak{v}$. The span of a radio $k$-coloring $\mathfrak{h}$ is the largest integer assigned by $\mathfrak{h}$. The radio $k$-chromatic number of $\mathcal{G}$, denoted $rc_k(\mathcal{G})$, is the minimum span among all possible radio $k$-colorings of $\mathcal{G}$. For $k=d-1$, the radio $k$-coloring is known as an antipodal coloring, and the number $rc_{d-1}(\mathcal{G})$ is called the antipodal number of $\mathcal{G}$, also denoted by $ac(\mathcal{G})$. In this work, we derive the antipodal number of graphs having triameter less than or equal to $2d+1$, or equal to $3d$ allowing a particular vertex ordering with respect to distances. Then, applying these results and obtain the antipodal number for the Generalized Petersen graphs $GP(4\mathfrak{t}+2,2)$, $GP(4\mathfrak{t},2)$ for $\mathfrak{t}\geq3$ and Hamming graphs $H(3,3)$ and $H(4,3)$.
Consider a closed Riemannian manifold $M$ where the metrics $g(t)$ evolve by the Ricci-Bourguignon flow (RBF).The paper first derives a Sobolev inequality along the RBF. Then, by applying this Sobolev inequality we obtain a linear parabolic estimate and an upper bound estimate for the heat kernel along the RBF. Additionally, it confidently investigates the $\kappa$-noncollapsing estimates along the RBF.
