In this paper, a comprehensive study of the fuzzification and defuzzification of the Schrödinger equation in fuzzy neutrosophic environment is presented involving fuzzy truth (T) component, an indeterminacy (I) component, and a falsehood (F) component. Moreover, a modified numerical method of second-order accuracy in both time and space based on Crank–Nicolson method is reformulated and implemented to solve the neutrosophic Schrödinger equation under both amplitude and phase terms. The triangular neutrosophic number t was used. The stability of the modified Crank–Nicolson method has been investigated using the von Neumann method to show that the proposed approach is unconditionally stable. A numerical experiment is carried out, and the results obtained indicate the effectiveness and reliability of the proposed modified approach. Finally, an accuracy study comparing the neutrosophic exact and numerical results at the α,β,γ-cut level set is conducted. The mathematical novelty of this work lies in reformulating the classical Crank–Nicolson finite difference scheme, for the first time, into six coupled systems that govern the lower and upper bounds of all three neutrosophic components of the neutrosophic fuzzy Schrödinger equation (NFSHE) simultaneously, and in extending the von Neumann stability criterion to prove unconditional stability of the NFSHE across this coupled neutrosophic system, a result with no counterpart in the crisp or single-valued fuzzy literature.
Motivated by the mathematical modeling of heterogeneous viscoelastic media with nonstandard damping mechanisms, this paper investigates a Kirchhoff‐type viscoelastic wave equation involving a variable‐exponent nonlinear damping term, a logarithmic source, and a memory kernel. The main novelty of the work lies in the simultaneous treatment of three nonlinear effects: the nonlocal Kirchhoff coefficient, the viscoelastic memory term, and the spatially dependent damping exponent. This combination creates several analytical difficulties, mainly due to the loss of homogeneity in variable‐exponent spaces, the singular nature of the logarithmic source near the origin, and the interaction between the memory term and the Kirchhoff nonlinearity. Under suitable assumptions on the variable exponents and the relaxation kernel, we first establish the associated energy identity and show that the energy is nonincreasing. Then, by using potential‐well‐type arguments, Sobolev embeddings, and suitable auxiliary functionals, we prove the global existence and boundedness of solutions corresponding to initial data in the stable set. On the other hand, for initial data with negative initial energy, we construct a perturbed functional of Levine type and derive a differential inequality which implies finite‐time blowup. The results provide a clear dichotomy between global existence and finite‐time blowup for a Kirchhoff‐type viscoelastic equation with logarithmic nonlinearity and variable‐exponent damping.
Let G be a graph with n vertices and Laplacian eigenvalues μ1,μ2,…,μn. The Laplacian Estrada index of G is defined as LEEG=eμ1+⋯+eμn. In this paper, using the Karush–Kuhn–Tucker optimization framework under inequality constraints, we establish new lower bounds for LEEG in terms of the two largest degrees of G. Specifically, if G is a connected graph with n≥3 vertices, m edges, and degree sequence d1≥d2≥⋯≥dn, then LEEG≥1+ed1+1+ed2+n−3e2m−d1−d2−1/n−3 if d3+⋯+dn≤n−3d2 and LEEG≥1+ed1+1+n−2ed2 otherwise. These inequalities improve previously known results on the Laplacian Estrada index.
The main focus of this manuscript is to provide generalizations of logarithmic mean and extended beta function in the framework of matrix setting. Some properties such as partial derivative formulas, functional relations, inequalities, integral representations, finite sums, and infinite sums are obtained for this new function of matrix arguments. Also, an application of these functions of probability theory is demonstrated.
Suppose that G is a connected graph. For any two vertices u and v, let dG (u,v) denote the distance between u and v in G. The diameter of G is the maximum distance between any pair of vertices, and it is denoted by diamG. A multilevel distance labeling (or radio condition) for G is a function f that assigns to each vertex of G a positive integer such that for any distinct vertices u and v, du,v+fu−fv≥diamG+1. The largest positive integer in the range of f is called the span of f. The radio number of G is denoted by rnG, is the minimum span of a multilevel distance labeling for G. In this paper the multilevel distance labeling of Hn is calculated for every n≥15: rnHn=4n+2.
In this work, a nonlinear deterministic mathematical model is proposed to explore the evolution of cardiac tissue behavior from healthy to at-risk, to damaged, to treated, and to recovered and the consequences of these changes on the dynamics of electrocardiogram (ECG) signals. The model seeks to set up a single solution to understand how diseases develop, how the nonlinear tissue laws mimic this behavior, and how treatments affect the outcome. Analytical investigations determine the positivity and boundedness of solutions, and disease-free and endemic equilibrium states are detected. The basic reproduction number is obtained as a threshold parameter that regulates cardiac stress persistence. Using bifurcation and stability analysis, the authors find transcritical, backward, and Hopf bifurcations; by changing the stress transmission and treatment effectiveness, cardiac tissue and ECG dynamics change. The partial rank correlation coefficients (PRCC) approach is used to conduct the global sensitivity analysis that determines the most influential parameters affecting the disease progression and long-term system behavior, which is illustrated by numerical simulations showing the nonlinear transition of the system. Moreover, an optimal control scheme is studied to assess therapeutic strategies that decrease damage and improve healing in the tissues. Their findings offer a theory in favor of myocardial progression of cardiac disease and the value of effective treatment interventions for stabilizing the heart and improving recovery outcomes.
For an integer k≥2, a spanning k-hypertree T is defined as a spanning hypertree (β-acyclic) such that the maximum degree dTv of every vertex v∈VT is at most k. A sufficient condition for the existence of a spanning k-hypertree in connected hypergraphs is established. The algorithm for checking the existence of a spanning k-hypertree and its complexity analysis are also presented. The distance spectral radius of a hypergraph H, denoted as λDH, is defined as the largest eigenvalue of its distance matrix DH. A lower bound for λDH is established for a connected hypergraph H. Combined with typical distance spectral techniques and structural analysis of hypergraphs, this bound yields sufficient conditions for the existence of spanning k-hypertrees in terms of the distance spectral radius.
In this paper, we apply the similarity method to a nonlinear recursive sequence of order eight, where the parameters are nonzero real number sequences with real initial values. We derive Lie symmetry generators of the recursive sequence and derive a formula to its well‐defined solutions. We study the global behavior of its solutions, and finally, we introduce the forbidden set of that recursive sequence.
We define a new convergence concept called convergence in ψ ‐density. Let be the class of strictly increasing, differentiable, and unbounded functions ψ : (0, ∞ )⟶(0, ∞ ). We say that a sequence x = { x k } is convergent in ψ ‐density to L if the limit , where B ≔ B ( ϵ )≔{ k ≤ i : | x k − L | ≥ ϵ } for every ϵ > 0. If the function is concave, our convergence method strictly implies asymptotic density convergence, which is widely recognized as statistical convergence. In this paper, we first establish the fundamental properties of ψ ‐density convergence. Subsequently, we prove a Korovkin‐type approximation theorem for sequences of positive linear operators (pLOs) under this new framework. We provide an illustrative example, supported by graphical representations, to verify our theoretical findings. Finally, we compute the rate of convergence of these operators in terms of the modulus of continuity.
Let R be a commutative ring with unity. The Jacobson graph of R is an undirected simple graph whose vertex set is R \ J ( R ), where J ( R ) is a Jacobson ideal of R , and for any distinct vertices x , y ∈ R \ J ( R ) are adjacent if and only if 1 − x y is not a unit of R . In this paper, we compute the degree‐based topological indices of the Jacobson graph such as the first and second Zagreb Indices, Product Connectivity Index, Sum Connectivity Index, Atom‐Bond‐Connectivity Index, Randic Index, Sombor Index, Geometric–Arithmetic Index and the Arithmetic–Geometric Index.
Let G be a connected graph. A subset S⊆VG is called a general position set of a graph G if no shortest path in G contains more than two vertices from S. Likewise, a subset X⊆EG is an edge general position set if no shortest path contains more than two edges from X. The maximum cardinalities of such sets are known as the general position number and the edge general position number, respectively. This paper focuses on determining these values for the unitary Cayley graph GZn, employing structural methods based on the direct product of graphs. For n=p1α1p2α2 with p1
This work discusses the time-tempered fractional Burgers’ equation in the context of Caputo tempered fractional derivative. This equation is crucial for modeling non-Newtonian fluid flow and long-memory phenomena that involves a time-tempering effect. In this paper, we employ a novel approach that combines Laplace transformation and the residual power series method to obtain approximate analytical solutions for the time-tempered fractional Burgers’ equation, the tempered Laplace residual power series method. The proposed method is not an incremental extension of existing Laplace-based approaches; it requires a fundamentally new expansion structure and original convergence theory and yields solutions that encode the tempering dissipation effect intrinsically through a closed-form exponential factor. The results obtained during the work are presented in two- and three-dimensional figures, in addition to tables showing the exact numerical results for comparison and a study of the relative and absolute errors of these solutions. In this paper, we explore the impact of the fractional derivative on the solutions obtained through our method and provide an explanation for these effects. We present the results at various parameter values in the governing equation to examine the resulting changes in these parameters. We also show real-world examples related to the time-tempered fractional Burgers’ equation to demonstrate how useful this study is for modeling complex patterns. The modeling of intricate fluid mechanics phenomena and the creation of novel numerical algorithms to control these models are made possible by this work. Additionally, it illustrates the significance of fractional differential equations in several scientific domains, including engineering and physics.
The adaptive progressive first-failure censoring scheme is a flexible life-testing design that integrates the efficiency of first-failure experiments with the adaptability of progressive censoring, ensuring the observation of a predetermined number of failures while allowing dynamic control over test duration and resource allocation. Despite its practical advantages in reliability experimentation, statistical inference under this censoring structure remains limited in the existing literature, particularly with respect to the estimation of reliability measures. Motivated by this gap, this paper develops classical and Bayesian inference for the XLindley lifetime distribution under the adaptive progressive first-failure censoring scheme. The XLindley distribution, characterized by a single positive parameter, provides sufficient flexibility to model positively skewed lifetime data while maintaining analytical tractability under complex censoring mechanisms. Maximum likelihood estimation is derived for the model parameter as well as for the reliability and hazard rate functions, and two types of approximate confidence intervals are constructed based on normal approximation and log-transformed estimators. Bayesian estimation is implemented under the squared error loss function using Markov chain Monte Carlo simulation, from which symmetric Bayesian credible intervals and highest posterior density intervals are obtained. The finite-sample behavior of the proposed estimators is examined through an extensive Monte Carlo simulation study. The methodology is further illustrated through the analysis of two real datasets, including survival times of gastric cancer patients receiving combined chemotherapy and radiation therapy, and monthly rainfall measurements recorded in New South Wales, Australia. The results demonstrate the flexibility of the XLindley model and the effectiveness of the proposed inferential procedures under adaptive progressive first-failure censoring.
Pseudorandom sequences with large linear complexity are widely employed in practical secure communication systems, such as stream ciphers, spread-spectrum communications, and wireless networks. They play a critical role in enhancing resistance against linear cryptanalysis. Motivated by practical cryptographic requirements, in this work, we investigate binary and r-ary sequences derived from Euler quotients modulo pmqn, where p and q are distinct odd prime numbers. Under the condition that gcdpq,p−1q−1=1, we construct a binary sequence with period pm+1qn+1 and determine its linear complexity. Furthermore, according to the ring theory of residue classes, we construct a new class of binary sequence with period pmqn+1 when p divides q−1. By analyzing the roots of the characteristic polynomial of this sequence over F2, the linear complexity of the sequence is obtained. To extend the construction for broader cryptographic deployment, we generalize the binary sequence of period pmqn+1 to an r-ary sequence for an odd prime r and present its linear complexity under the conditions r∤p−1 and rq−1≢1 modq2. It is shown that the linear complexity of these sequences is at least half of their period, implying strong resistance to the Berlekamp–Massey algorithm. The proposed sequences are suitable for stream cipher design, secure random number generation, and other communication security engineering scenarios that require long period, high linear complexity pseudorandom signals. This is an open access article under the terms of the Creative Commons Attribution-Noncommercial License, which permits use, distribution, and reproduction in any medium, provided that the original work is properly cited and is not used for commercial purposes.
Rainfall-deficit derivatives provide financial protection against precipitation shortfalls in agriculture, water management, and other weather-sensitive activities. Their valuation is complicated by the seasonality, local dependence, and nontradability of rainfall. An inadequate dispersion specification may produce intervals that are too narrow in highly variable months or unnecessarily wide in relatively stable months. We introduce a seasonal mean-reverting uncertain rainfall-index model with month-dependent uncertainty intensity. We derive an explicit integral representation of its α-paths, prove that the associated cumulative rainfall-deficit functional is decreasing in the belief level, and obtain its inverse uncertainty distribution. These results reduce the valuation of a put on the cumulative deficit index to a one-dimensional deterministic integral. We also derive the exact monthly transition of the centered process, which provides the basis for parameter estimation. We calibrate the model to monthly observations from Berlin–Tempelhof by estimating the seasonal mean through a Fourier representation and the transition parameters from the centered series. Predictive performance is evaluated over a 1996–2005 holdout period against a matched Gaussian Ornstein–Uhlenbeck benchmark. The two models have nearly identical monthly point-forecast accuracy. The uncertain model yields slightly lower monthly interval scores and quantile losses and performs marginally better for the 1-month June deficit contract. The Gaussian benchmark performs better for the April–June accumulation contract, for which the uncertain model produces substantially wider predictive distributions. The uncertain model therefore provides a competitive short-horizon specification, while its excessive multimonth dispersion limits its suitability for longer accumulation contracts.
We develop a finite difference method for time-fractional diffusion equations involving the Atangana–Baleanu–Caputo (ABC) derivative with a nonsingular Mittag–Leffler kernel. The temporal derivative is approximated by an ABC-adapted L1-type formula, while the spatial derivative is discretized by a second-order central difference scheme. The resulting fully discrete system is analyzed using the positivity and monotonicity of the ABC memory weights together with an M-matrix argument. The scheme is shown to be unconditionally stable and convergent with accuracy Oτ2−α+h2. Numerical experiments confirm the theoretical convergence behavior and illustrate the effect of the fractional order on the diffusion dynamics.
This paper presents the first systematic analysis of critical exponents for both third and fourth-order fractional Moore–Gibson–Thompson (MGT) equations with combined spatiotemporal fractional derivatives and structural damping. We establish sharp nonexistence results for global solutions via an adapted test function method, deriving explicit critical exponent formulas that quantify the interplay between fractional orders α,β,γ,δ, spatial dimension N, and the nonlinearity exponent k. Notably, our framework accommodates the nonmonotone source term ψk1−ψϑ, extending prior results for integer-order MGT and fractional wave equations.
This paper investigates the Hyers–Ulam stability of a class of generalized biquadratic functional equations involving four unknown mappings in non-Archimedean normed spaces. We analyze a coupled system that simultaneously incorporates additive, quadratic, additive–quadratic, and quadratic–additive behaviors, thereby capturing both symmetric and asymmetric interactions among the variables. By exploiting the ultrametric structure inherent in non-Archimedean normed spaces, we establish the existence and uniqueness of approximating solutions, together with stability bounds. Furthermore, a known stability result for biquadratic functional equations is recovered as a corollary of the main theorem.
In this paper, we introduce refined notions related to convergence for sequences of functions. These include relative alpha convergence and relative continuity at a point with respect to a scale function, as localized and strict parametric variants of classical alpha convergence. Furthermore, using these new concepts, we investigate their properties and relationships among these different notions. Lastly, we give a Korovkin-type theorem for relative alpha convergence; it will be observed that the conditions in the classical Korovkin theorem can be effectively extended.
The study of geometric structures on homogeneous pseudo-Riemannian spaces G/H,g has long been an important area of research in differential geometry. Such spaces are characterized by the transitive action of a Lie group G, with the metric g being G-invariant using mathematical operators. However, the work of Fels and Renner shows that four-dimensional homogeneous pseudo-Riemannian manifolds also include nonreductive cases, where the usual reductive decomposition does not hold. These nonreductive spaces provide a rich framework for investigating various geometric structures and nonlinear systems. In this paper, we study hyperbolic Yamabe soliton vector fields, a concept introduced only recently.
