Let R be a non-commutative ring with 1 ≠ 0 and S(R) be the set of all ideals of R. In this paper, we extend the concept of 2-absorbing principally right primary ideals to the context of 𝜙-2-absorbing principally right primary ideals. Let 𝜙 : S(R) → S(R) ∪ {Ø} be a function. A proper ideal I of R is said to be a 𝜙-2-absorbing principally right primary ideal of R if whenever a, b, c ∈ R with aRbRc ⊆ I and aRbRc ⊈ 𝜙(I) implies ab ∈ I or ac ∈ $\sqrt{I}$ or bc ∈ $\sqrt{I}$. A number of results and characterizations concerning 𝜙-2-absorbing principally right primary ideals are given.

The authors in [3] have defined a (commutative unital) ring R to be S-coherent if each finitely generated ideal of R is an S-finitely presented R-module. Any coherent ring and any S-Noetherian ring is a S-coherent ring. In this paper, we investigate the transfer of the property of S-coherence to the amalgamation A ⋈^f J. We provide necessary and sufficient conditions for A ⋈^f J to be an S-coherent ring.

Let R be a commutative ring with identity and S a multiplicative subset of R. This article is motivated by the results of Mimouni on Q-ideals. In this paper, we introduce and investigate the notion of S-Q-ideals which is a generalization of Q-ideals introduced by Mimouni in [8]. Let I ⊆ Q be two ideals of R with Q ∩ S = Ø. The ideal I is said to be an S-Q-ideal if there exists s ∈ S such that whenever ab ∈ I with a, b ∈ R, then sa ∈ I or sb ∈ Q. In Section 2, we study some properties of S-Q-ideals. Among other results, we give some characterization of the S-Q-ideal of the idealization of a module. On the other hand, we show I is an S-Q-ideal of R if and only if I[X] is an S-Q[X]-ideal of R[X], provided that R is a domain and S is anti-Archimedean. In Section 3, we introduce a weak version of S-Q-ideals and we study some of their properties.

The purpose of this paper is to prove that every local and 2-local derivation in dual quaternion algebra is a derivation. Also, we determine all biderivations, commuting linear maps, and the centroid of dual quaternion algebra 𝓗_d over the reals.

This paper focuses on the properties and factorizations of the Chebyshev polynomials. We show that the Chebyshev polynomial s_n(x) with the particular initial conditions s₀(x) = 0, s₁(x) = 1 has a simple factorization, and each factor in this factorization has a one-to-one correspondence with a factor of the integer n. More precisely, we define a family of polynomials P_k(x) with integer coefficients for each k = 1, 2, …, and show that they satisfy s_n(x) = ∏_d|n P_d(x) for all n = 1, 2, …. We also present some more results about the factorization properties and the irreducibility properties for the Chebyshev polynomials. Additionally, we establish a characterization for certain irreducible polynomials, including the Chebyshev polynomial v_n(x) with the particular initial conditions v₀(x) = 1, v₁(x) = x - 1. We demonstrate that any nonconstant polynomial g(x) with integer coefficients that satisfies a specific identity, g(x)g(-x) = ±g(±(2-x²)), can be expressed as a product of irreducible polynomials Ψ_k(x), where Ψ_k is a polynomial such that ${\Psi}_k(x\,+\,{\frac{1}{x}})\,=\,x^{-{\frac{1}{2}}{\varphi}(k)}{\Phi}_k(x)$. (Here 𝚽_k is the k-th cyclotomic polynomial and φ is Euler's totient function.)

Suppose that 𝕽 represents a prime ring of characteristic different from 2 with its right Martindale quotient ring 𝕯, extended centroid 𝕮 and f(x₁, ..., x_n) symbolizes a noncentral multilinear polynomial over 𝕮. Let 𝕵 be a nonzero ideal of 𝕽 and b ∈ 𝕯. If 𝕱, 𝕲 and 𝕳 are three b-generalized derivations of 𝕽 such that 𝕱(f(s))f(s) + 𝕲(f(s))f(s)v = 𝕶(f(s)²) for all s = (s₁, ..., s_n) ∈ 𝕵ⁿ and for some v ∈ 𝕹, then we elaborate all probable constructions of the maps.

In this paper, we establish sufficient conditions for the existence of solutions to the set-valued equilibrium problem without requiring the closure of the constraint set. Additionally, the conditions for the semicontinuity and convexity of the objective function are only needed on a special subset of the constraint set, rather than covering its entire domain. As applications, we derive existence results for minimax problems on dense sets.

In this paper, we introduce a novel subclass of bi-univalent functions associated with Lucas-Balancing numbers in the unit disk. Precise estimates are derived for the Fekete-Szegö functional and the Taylor-Maclaurin coefficients within this new class. The results are further analyzed for specific parameter values, showcasing the applicability of the proposed definitions and proofs. These findings not only contribute to the study of bi-univalent function subclasses but also open new avenues for exploring related classes in analytic function theory.

Dini's theorem is examined in the context of discontinuous functions, particularly semi-continuous functions. It is further demonstrated that uniform convergence is equivalent to the Hausdorff convergence of graphs in the product space as a result of applying Dini's theorem for semi-continuous functions.

The Miao-Tam equation gives an Einstein metric but the Fischer-Marsden equation does not, when an associated function to these equations is a non-zero constant. In this article, the critical equation 𝓛^*_g(λ) = αg is introduced that generalizes the famous Miao-Tam equation and the Fischer-Marsden equation and gives an Einstein metric when the associated function is a non-zero constant. It is shown that a Sasakian 3-manifold (M, g) admitting a non-constant solution of the equation 𝓛^*_g(λ) = αg is Einstein and has constant curvature 1. In addition, if (M, g) is complete, then it is compact Einstein and isometric to a unit sphere. Several consequences are drawn from these results. Finally, an example is constructed to verify the results.

In this paper we discuss m-quasi-Einstein metric on three dimensional trans-para-Sasakian manifold whose potential vector field is pointwise collinear with Reeb vector field. Next we prove conditions for para-Sasakian manifold with to be para-cosymplectic.

The aim of this study is to characterize the 𝜼-Ricci-Bourguignon soliton (briefly, 𝜼-𝓡𝓑𝓢) in the context of 𝝃-conformally flat Lorentzian para-Kenmotsu manifold. Further, we establish that the scalar curvature 𝜿 of a p-dimensional Lorentzian para-Kenmotsu (briefly, (𝓛𝓟𝓚)_p) manifold admitting an 𝜼-𝓡𝓑𝓢 satisfies the Poisson's equation. Next, the condition for which the scalar curvature of (𝓛𝓟𝓚)_p manifold admitting an 𝜼-𝓡𝓑𝓢 establishes Laplace equation. The presence of gradient 𝜼-𝓡𝓑𝓢 on (𝓛𝓟𝓚)_p manifold has been established. At the end, an illustration of an (𝓛𝓟𝓚)₄ manifold is constructed to confirm the results of this manuscript.

This paper introduces and examines a new type of Riemannian manifold called a weakly Schouten symmetric manifold, referred to as (W SS)_n. We establish the existence of such a manifold through a concrete example. We also explore various geometric properties of (W SS)_n under specific curvature conditions and proved sufficient criteria for these manifolds to be classified as Einstein manifolds. Finally, we analyze the properties of decomposable weakly Schouten symmetric manifolds.

In this article, we introduce an indefinite Kenmotsu statistical manifold and study its screen semi-invariant lightlike submanifolds. We investigate the integrability of distributions, self adjointness of shape operator and symmetricity of second fundamental forms of screen semi-invariant lightlike submanifolds of an indefinite Kenmotsu statistical manifold. We also provide two examples to illustrate these concepts.

We introduce the concept of m-transitivity and study some of the properties and the relation with some other types of generalizations of transitivity. We prove that the m-transitivity is a unified form of the classical transitivity and many other types of transitivity such as 𝛿-transitive, 𝛼-transitive, b-transitive, and e^*-transitive.

We define the concept of stronger forms of positively expansive maps and name them p𝓕-expansive maps. Here 𝓕 is a family of subsets of ℕ. Examples of p𝓕-expansive maps, for 𝓕 = 𝓕_t, 𝓕_s, 𝓕_cf, are discussed. Also, we give examples of maps which are p𝓕_t-expansive but not p𝓕_cf-expansive, p𝓕_s-expansive but not p𝓕_t-expansive, and p𝓕_s-expansive but not p𝓕_cf-expansive. Further, we obtain conditions under which a positively expansive map is p𝓕_cf-expansive and p𝓕_s-expansive map. We next study several properties of p𝓕-expansive maps. A characterization of p𝓕-expansive maps in terms of 𝓕^*-generator is obtained. Here 𝓕^* is dual of 𝓕. Considering (ℤ, +) as a semigroup, we study 𝓕-expansive homeomorphism, where 𝓕 is a family of subsets of ℤ \ {0}. We show that there does not exist expansive homeomorphism on a compact metric space which is 𝓕_s-expansive. Also, we study relation between 𝓕-expansivity of f and the shift map σ_f on the inverse limit space.

In this paper, we provide sufficient conditions under which any locally isometric covering semi-conjugation between two dynamical systems preserves various forms of expansivity and shadowing property. We also give necessary and sufficient conditions under which the orbital limit of an orbitally convergent sequence of functions admits various forms of expansivity and shadowing property.

This article concerns the existence of infinitely many fast homoclinic solutions for the following damped vibration system with p-Laplacian $$\frac{d}{dt}\({\mid}{\dot{u}}(t){\mid}^{p-2}\,{\dot{u}}(t)\)\,+\,q(t){\mid}{\dot{u}}(t){\mid}^{p-2}\,{\dot{u}}(t)\\-V(t){\mid}u(t){\mid}^{p-2}\,u(t)\,+\,{\nabla}W(t,\,u(t))=0$$ where p ≥ 2, q, V ∈ C(ℝ, ℝ) and W ∈ C¹ (ℝ × ℝ^N, ℝ). Applying Fountain Theorem and Dual Fountain Theorem, we prove that above system possesses two different sequences of fast homoclinic solutions when V satisfies a new coercive condition and the potential W(t, x) is a combination of a superquadratic and a subquadratic functions.

The generalized Sierpiński graph is a well-known class of interconnection network. Any network can be modeled by a connected graph G = (V, E), where V is the set of nodes and E is the set of the direct communication links between the nodes. Many graph theoretical parameters are useful in the study of the efficiency and reliability of an interconnection network model. Let G = (V, E) be a graph, and u, v ∈ V . Then, u strongly dominates v if (i) uv ∈ E and (ii) d(u) ≥ d(v). A set D ⊆ V is a strong-dominating set (sd-set) of G if every vertex in V \ D is strongly dominated by at least one vertex in D. The strong domination number 𝛾_st of G is the cardinality of the smallest sd-set. In this paper, we obtain an upper bound for the strong domination number of the generalized Sierpiński graphs. We obtain a condition for the graphs to attain the bound.

In this paper, we obtain new sufficient conditions for the existence and uniqueness of solutions of Caputo fractional linear differential equations, via Banach fixed point theorem. Then, we investigate Hyers-Ulam and Hyers-Ulam-Rassias stability for such equations. Finally, two illustrative examples are given to show the applicability of the theoretical results.