In this work, we introduce a generalized class of fractional Dirac operators constructed through the Whittaker transform of the Mittag–Leffler function using the generalized Fox–Wright function. The framework extends the theory of Dirac operators to the fractional setting on Riemannian manifolds via Clifford algebras, spin representations, and the geometry of fiber bundles. Fundamental properties of these operators are established, showing that they combine conventional and fractional derivatives, thereby generalizing key results in Riemannian geometry. In particular, we extend the Schrödinger–Lichnerowicz formula to fractional order, revealing new scalar curvature terms governed by different fractional derivatives. This approach bridges fractional calculus and geometric analysis, offering potential applications in field theory and the study of manifolds with fractal dimensions. Furthermore, we analyze the fractional Dirac operator on Lorentzian cylinders under Atiyah–Patodi–Singer boundary conditions, deriving corresponding constraints on the fractional exponents. The formulation naturally leads to a fractional Schrödinger–Dirac equation encompassing higher-order derivative terms. Our findings open prospects for extending the fractional formalism to spectral geometry and noncommutative spaces, particularly through the development of a fractional spectral action principle, where the geometric action is constructed from the spectrum of a fractional Dirac operator, and the investigation of index theorems and algebraic geometric structures associated with fractional Dirac operators, which generalize the classical Dirac operator by incorporating fractional-order nonlocal effects.
This paper studies the left-definite theory of dynamic Sturm–Liouville problems in the regular case.
Picture fuzzy sets as a generalization of traditional and intuitionistic fuzzy sets, offer a powerful mathematical framework for modeling uncertainty and imprecision in real-world situations. By incorporating positive, neutral, and negative membership degrees, PFSs provide a more comprehensive representation of human reasoning and decision preferences. This study introduces novel picture fuzzy topological operators for interior and closure, derived from standard picture fuzzy modal operators, and develops a compactification approach for picture fuzzy topological spaces based on picture fuzzy filters. Fundamental topological concepts such as compactness, regularity, normality, and the Hausdorff condition are analyzed within this framework. Furthermore, an efficient decision-making algorithm is proposed to extend the theoretical model to practical applications. The effectiveness of the proposed approach is illustrated through a real-world case study on the 2026 Tamil Nadu election, where picture fuzzy compact spaces are used to model voter preferences and evaluate political candidates in a transparent and unbiased manner. The results demonstrate the potential of picture fuzzy topology as a rigorous decision-support tool that models uncertainty, and promotes equitable political evaluation.
In this study, we derive Ostrowski-type inequalities associated with the Riemann-Liouville type fractional Lupaş-Kantorovich operators (for short RLFLK operators), created by merging the Lupaş-Kantorovich operator with the Riemann-Liouville fractional integral operator. The main tools of our approach are the notions of convexity and boundedness. Based on this framework, we establish several integral inequalities linked to the Ostrowski-type results for RLFLK operators. Moreover, we provide a key integral identity involving RLFLK operators on differentiable functions, which forms the basis of our method. We also support our theoretical findings with numerical examples and graphical illustrations. To the best of our knowledge, these results represent the first systematic investigation of integral inequalities in the framework of RLFLK operators.
In this paper, we introduce the concepts of neutrosophic Catalan I-convergent sequence space ℛ_(𝒢)^I(𝒮) , neutrosophic Catalan I-null convergent sequence space ℛ_(𝒢)^I^0(𝒮) , and neutrosophic Catalan I-bounded sequence space ℛ_(𝒢)^I^∞(𝒮) in neutrosophic normed spaces by using the notions of convergent sequences (c), null sequences (c_0) , bounded sequences (l_∞) , and the domain of the regular Catalan matrix 𝒮=(b_nk) . Furthermore, we investigate the concept of neutrosophic Catalan I-Cauchy sequences in neutrosophic normed spaces. We also define open balls ℬ_x^I(t,ϵ )(𝒮) and closed balls ℬ_x^I[t,ϵ ](𝒮) , and study several algebraic and topological properties of these newly defined sequence spaces.
In this paper, we present new inner product inequalities that can be viewed as Cauchy-Schwarz-type or Buzano-type inequalities. We then apply these inequalities to derive applications for the numerical radius and the Euclidean operator radius of two operators. The obtained results are compared with many existing results in the literature in a way that supports the advantage of the new findings.
Given a closed Riemannian manifold (M, g) of dimension n≥ 3 , we investigate the existence of a nodal solution to the equation P_g^ku=f|u|^2^♯-2u+λ |u|^q-2u, where P_g^k is a coercive polyharmonic operator, f is a smooth positive function on M, q∈ (1,2) , λ∈ (0,+∞ ) and 2^♯:=2n/n-2k is the critical Sobolev exponent for the embedding H_k^2(M)↪ L^2^♯(M). Using variational techniques, critical points theory and by applying a very nice minimization relies on a suitable constraint. Supposing that |Isom(M,g)|≥ 2 and for λ >0 and small, we prove the existence of a weak nodal solution in H_k^2(M) .
The main objective of this paper is twofold. First, we provide a complete proof of a fixed point theorem for forward ϕ G -contractions in quasi-metric spaces, thereby filling a gap in the recent work of Zakiyudin et al. Second, we apply this theorem to investigate the existence and uniqueness of mild solutions to a nonhomogeneous Cauchy problem. To this end, we introduce a suitable quasi-metric structure on an appropriate function space and show that the associated solution operator satisfies the assumptions of the fixed-point theorem. Consequently, the existence and uniqueness of mild solutions are obtained as an application of the developed theory. Our results generalize previous applications of ϕ G -contractions to the heat equation and illustrate the effectiveness of fixed-point methods in the study of nonhomogeneous evolution equations.
Let R be a commutative ring with non-zero identity. The Jacobson graph of R denoted by 𝔍_R is a graph with the vertex set R∖ J(R) , and two distinct vertices x and y are adjacent if and only if 1-xy ∉ U(R) , where U(R) is the set of all unit elements of R. Let γ _ℜ(𝔍_R) be the Roman domination number of 𝔍_R . In this paper, we determine all Artinian commutative rings R with γ _ℜ(𝔍_R)≤ 6 .
In this article, we investigate generalized Ricci solitons on four-dimensional simply-connected non-symmetric pseudo-Riemannian generalized symmetric spaces and classify them up to isometry.
In this paper, we prove two estimates for the generalized Fourier transform in the space square integrable functions on certain classes of functions characterized by the continuity modulus.
The Hamming matrix H(G) of a graph G encodes the Hamming distances between the binary incidence strings of its vertices and gives rise to Hamming spectrum and Hamming energy HE(G). Building on recent foundational paper that established bounds for paths and closed-form formulas for regular graphs, we develop a unified approach via equitable partitions to determine the full Hamming spectrum and energy of several graph families, such as semi-regular bipartite graphs, complete multipartite graphs, wheel graphs, windmill graphs and the Dutch windmill graphs. In each of these graph classes we partition the vertex sets into a small number of natural classes so that the corresponding quotient matrix B is only 2× 2 (or 3× 3 in the Dutch windmill case). Diagonalizing these low-dimensional matrices yields explicit eigenvalues for H(G) and hence closed-form expressions for HE(G). Our results significantly extend the list of graphs for which Hamming-based spectral invariants are known in closed form, and illustrate the power of equitable partitions in spectral graph theory.
In this paper, we introduce the concept of the Gelfond-Leontiev Sălăgean-difference operator for an analytic function in the unit disc. This new operator and concept of subordination are used to define a new subclass of analytic univalent functions that consists of several known and new generalizations of bounded turning functions. Some coefficient bounds for this class are considered. Bounds of initial Taylor coefficients and logarithmic coefficients are established. Also, Fekete-Szegö functional and second Vandermonde determinants whose entries are both initial coefficient and logarithmic coefficients are proved.
This paper presents a systematic study of the mixed degenerate Gould–Hopper type polynomials, beginning with their generating functions and fundamental operational properties. Summation identities, addition formulas, and a determinant representation are developed to reveal the underlying algebraic structure. A computational investigation of the zero distributions is carried out, supported by numerical illustrations and pattern analysis. The results provide insight into the analytical behavior of the family and its structural richness. Concluding remarks discuss theoretical implications and potential directions including asymptotics, orthogonality, and applications in mathematical physics.
Drug toxicity and therapeutic effects are two concepts in pharmacokinetics that coexist in conflict. Too much or an insufficient amount of the drug in the body is not ideal. This brings about the need to optimise these two contrasting but important factors. As such, we developed a system of three ordinary differential equations to describe the dynamics of drug concentration, taking into account the effect of drug-metabolising enzymes and periodic intravenous injections, using the Dirac Delta function as the source term. We noted that frequent administration of drugs helps achieve the minimum level of drug efficacy. The model with a single dose has (0, 0, 0) as the only stable equilibrium point. However, when periodic injections are administered, this equilibrium is destroyed, and a new stable non-zero equilibrium point emerges.
In this paper, we consider a generalization of the complex Helton class to the multivariable setting. Inspired by the work [Al Rwaily, A., Higher order quasi complex Helton class of Hilbert space operators, Filomat 38 (31), 10819–10834 (2024) https:doi.org/10.2298/FIL2431819A], we introduce the complex Helton class of tuples of commuting operators. Given a positive integer m, for a conjugation C on a complex infinite-dimensional Hilbert space ℋ and for commuting d-tuples R=(R_1,… ,R_d), S=(S_1,… ,S_d) , we say that S belongs to the complex Helton class of R with order m, and we denote this by S∈ Helton_C,m(R), if ∑ _k=0^m(-1)^m-k m ()k (∑ _i=1^dR_i )^kC (∑ _i=1^dS_i )^m-kC=0. In the present work, some basic structural properties of such a class of a commuting tuples are established, especially the single valued extension property (SVEP) property and Bishop’s property (β ) . Then, we prove that the complex Helton class is invariant under nilpotent perturbation. Finally, spectral properties are shown.
Abstract A finite group G is called ( l , m , n ) -generated , if it is a quotient group of the triangle group $$T(l,m, n) = \left.$$ T ( l , m , n ) = x , y , z | x l = y m = z n = x y z = 1 . In [23], Moori posed the question of finding all the ( p , q , r ) triples, where $$p,\ q$$ p , q and r are prime numbers, such that a non-abelian finite simple group G is a ( p , q , r )-generated. In answering this question, we establish all the ( p , q , r )-generations for the group $$G_{2}(3).$$ G 2 ( 3 ) . We mainly used the structure constant method together with other results to establish the generation and non-generation of the $$G_{2}(3)$$ G 2 ( 3 ) by the triples ( p , q , r ). The Groups, Algorithms and Programming, GAP [21] and the Atlas of finite group representations [27] are used in our computations.
Let R be a principal ideal domain, and let ( 𝕋(V_n+1⊕ V_≤ n),∂ ) and (𝕋(W_n+1⊕ V_≤ n),δ ) be two free differential graded R -algebras such that ∂ = δ on v ∈ V_≤ n . This paper is dedicated to exploring the problem of constructing a DGA-map α :( 𝕋(V_n+1⊕ V_≤ n),∂ )→ (𝕋(W_n+1⊕ V_≤ n),δ ) such that the chain map α̃_* , induced by α on the indecomposables, satisfies α̃ = id on V_≤ n . Our focus is to provide an algebraic condition under which ( 𝕋(V_n+1⊕ V_≤ n),∂ ) and (𝕋(W_n+1⊕ V_≤ n),δ ) become quasi-isomorphic.
We investigate pseudo-almost periodic solutions for a class of nonautonomous difference equations in Banach spaces. Using discrete evolution family techniques combined with convolution operators, we establish existence and uniqueness results for linear systems driven by Stepanov pseudo-almost periodic forcing terms. These results are further extended to nonlinear equations under suitable Lipschitz conditions, ensuring the existence of unique pseudo-almost periodic mild solutions. As applications, we examine both a discrete heat equation with time-dependent coefficients and a discrete parabolic integrodifference equation with memory, demonstrating that pseudo-almost periodicity and Stepanov pseudo-almost periodicity are preserved in the long-term dynamics of these discrete systems.
