Let R = ⊕_α∈GR_α be a commutative ring graded by any group G. This paper explores the concept of graded strongly r-ideals in graded rings, expanding upon the understanding of strongly r-ideals in ungraded contexts. It delves into the fundamental properties of these ideals, establishing relationships with graded r-ideals and graded (1, r)-ideals. The research further investigates these ideals in specific types of graded rings, including graded trivial ring extensions. Key results include methods for constructing graded r-ideals that are not graded strongly r-ideals, and characterizing these in the context of product rings.

Let R be a commutative ring with an identity and M be a unitary R-module. For positive integers m and n, a proper submodule N of M is called an (m, n)-closed submodule if for r ∈ R and b ∈ M, r^mb ∈ N implies either rⁿ ∈ (N :_R M) or b ∈ N [22]. The purpose of this paper is to introduce the concept of weakly (m, n)-closed submodules as a generalization of (m, n)-closed submodules. A proper submodule N of M is called a weakly (m, n)-closed submodule if for r ∈ R and b ∈ M, 0 ≠ r^mb ∈ N implies either rⁿ ∈ (N :_R M) or b ∈ N. Many properties, examples and characterizations of weakly (m, n)-closed submodules are introduced, especially in multiplication modules. Moreover, we investigate the behavior of this structure under module homomorphisms, localizations, quotient modules, Cartesian product and idealizations. Finally, we define two kinds of submodules of the amalgamation module along an ideal and investigate conditions under which they are weakly (m, n)-closed submodules.

A ring R is said to be right 𝖈𝕻-Baer if the right annihilator of a cyclic projective right R-module in R is generated by an idempotent. This class of rings generalizes the class of right p.q.-Baer rings and abelian rings. We study the relationship between the 𝖈𝕻-Baer property of a ring R and those of the skew Hurwitz series ring R_H[[x; σ]]. We also study the necessary and sufficient conditions for the ring R_H[[x; σ]] to be APP.

Let I be a proper ideal of the commutative Noetherian ring R. One of the problems in local cohomology is to study the conditions under which cd(I, R) = 1. In this paper, we introduce a class of ideals, say ∆(R), that for each ideal I ∈ ∆(R), cd(I, R) = 1. We show that if there exists a finitely generated R-module N with Supp N = V (I) and pd_RN ≤ 1, then cd(I, R) ≤ 1. Also, for any I ∈ ∆(R) and R-module M with SuppM ⊆ V (I), M is I-cofinite if and only if Exⁱ_R(R/I, M) is finitely generated for i = 0, 1. Finally, I-transform functors enable us to give a necessary and sufficient condition for cd(I, R) = 1.

Let V be a hypersurface in ℙ⁴ with ordinary double and triple points as the only singularities, and let µ₂ and µ₃ be the number of ordinary double points (triple points, respectively) of V. Then we prove that if ${\mu}_2+11{\mu}m_3{\geq}{\frac{11d^4-50d^3+85d^2-70d+48}{24}}$, then the defect of V is positive.

In this note, we study the growth rate of the Engel digits and establish the Hausdorff dimensions of exceptional sets of points with a given relative growth rate.

In the present paper, under some new conditions we prove that the sum of two linear relations has the extended Bishop property in the extended complex plane. Also, we develop and generalize a local spectral theory for block matrix of linear relations. More precisely, this paper considers a necessary and sufficient conditions on the diagonal entries of 2 × 2-block matrix linear relations ensuring that this matrix has extended Bishop's property, extended Dunford's property and extended relatively single-valued extension property.

This paper will investigate the induced matrix norms of linear operators between finite dimensional vector spaces. In the past, the closed form formulas for ‖A‖_pq were known for some p, q = 1, 2, ∞, but not for ‖A‖_2∞ and ‖A‖_∞2. The formulas for these two norms and for ‖A‖₂₁ have been derived in this paper. The computation of ‖A‖₂₁, ‖A‖_∞1 and ‖A‖_∞2 is generally difficult, since they concern all the arrangements of objects. The algorithms developed here can provide the exact numerical value with the least amount of calculations.

The purpose of this paper essentially deals with the sequence spaces ℓ_∞(C_𝛼, Δⁿ, q), c(C_𝛼, Δⁿ, q) and c₀(C_𝛼, Δⁿ, q) defined as the domain of composition of Cesàro matrix C_𝛼 of order 𝛼 and the difference matrix Δⁿ of order n in the paranormed Maddox's sequence spaces ℓ_∞(q), c(q) and c₀(q), respectively. In addition, investigations have been performed to calculate their topological properties, inclusion relations, determine 𝛼-, 𝛽- and 𝛾- duals, and construct the Schauder basis of the new spaces. Finally, we emphasize the characterization of certain matrix mappings between the new described sequence spaces.

In this paper, we introduce the concept of a fuzzy *-double derivation and a fuzzy double derivation. Utilizing the fixed point method, we establish the Hyers-Ulam stability for the newly defined notions. Additionally, under specific conditions, we prove that every approximate fuzzy double derivation is, in fact, an exact fuzzy double derivation.

The primary focus of this article is to conduct a systematic study of a new class of Janowski-type q-symmetric harmonic p-valent functions, which are defined using the convolution operator combined with the Bell distribution series.

For -1 ≤ 𝛽 ≤ 1, ST[𝛽] is the subclass of generalized Pascal snail function defined by zf'(z)/f(z) ≺ 1/(1 - 𝛽z). In this article, we have presented the coefficient bounds for the analytic functions lying in the class ST[𝛽]. We have also presented the bounds for the Hankel determinants |H_2,1(f)| and |H_2,2(f)|. The bounds presented in this paper are sharp and the extremal function for each bound is also given.

This article introduces a class of possibly non-continuous operators, termed generalized Reich type enriched F-contractions, within the framework of normed spaces. These operators generalize existing Reich type contractive mappings and F-contractions, thereby broadening the scope of fixed point theory to encompass more general and nonlinear settings. The article establishes several fixed point and common fixed point theorems for this class, supported by illustrative examples. Importantly, these results are applicable to the study of Volterra integral equations and systems thereof, demonstrating the practical relevance of the theoretical framework. The developed approach not only enhances the understanding of nonlinear operator behavior but also contributes to the mathematical tools available for solving integral equations that arise in applied contexts.

In this paper, we use group algebras and semigroup algebras to provide counterexamples that demonstrate the failure of [11, Theorem 2.1, Proposition 3.2]. Additionally, we revisit and refine their results. Specifically, we establish that if A is a 𝜑-biprojective Banach algebra with a closed ideal I such that ${\bar{AI}}=I$, then A/I is ${\tilde{\varphi}}$-biprojective. Lastly, we explore the connection between 𝜑-approximate Helemskii biflatness and 𝜑-pseudo amenability.

In this paper, we construct conformal minimal two-spheres with constant curvature in ℍ𝑷⁴ by the twistor map π : ℂ𝑷⁹ → ℍ𝑷⁴. We show that, for 𝜙1, 𝜙2 and 𝜙3 in the Veronese sequence in ℂ𝑷⁹ , there is a unique one-parameter family of horizontal conformal minimal two-spheres with constant curvature $\frac{4}{25}$ , $\frac{4}{37}$ and $\frac{4}{45}$ respectively in ℍ𝑷⁴. In addition, using the holomorphic and anti-holomorphic Veronese two-spheres in ℂ𝑷⁸ and ℂ𝑷⁹, we construct two families of conformal minimal two-spheres in ℍ𝑷⁴ with constant curvature $\frac{1}{2}$ and $\frac{4}{9}$ respectively. Moreover, the conformal minimal two-spheres constructed in each family are not symplectic equivalent to each other.

In this paper, we demonstrate the existence of a unique left-invariant Riemannian metric on the real hyperbolic space of dimension n, up to automorphism. We derive the expression for the Ricci operator associated with this metric and show that it defines both an Einstein metric and an algebraic Ricci soliton. Finally, we examine the isometry group of the metric on the real hyperbolic space.

The objective of this paper is to explore some curvature properties of 𝓦₉-curvature tensor in Lorentzian 𝛼-Sasakian manifold (𝓜_𝛼). Initially, we establish certain vanishing properties of 𝓦₉-curvature tensor in 𝓜_𝛼 and arrive at a range of valuable insights. Subsequently, we describe 𝜑-𝓦₉-semi-symmetric condition in 𝓜_𝛼. Further, we explore 𝓜_𝛼 satisfying the condition 𝓠·𝓦₉ = 0. Again, we discuss 𝓦₉ Ricci pseudo-symmetric condition on 𝓜_𝛼. Next, we deal with 𝓜_𝛼 satisfying 𝓦₉·𝓡 = 0 condition. Later, we formulate 𝒱^♭-Ricci soliton and conformal 𝒱^♭-Ricci soliton. At last, we present an example of 5-dimensional 𝓜_𝛼 which verifies our results.

The bridge index is a useful invariant of knots. For classical knots, there is no ambiguity in its definition. For virtual knots, however, multiple natural definitions of the bridge index exist. To demonstrate that the difference between the two formulations can be arbitrarily large, we introduce a quantity called the semi-Wirtinger number of a virtual knot. A function of our numerical complexity turns out to also provide an upper bound for the number of biquandle homomorphisms from fundamental biquandles to other biquandles and offers a lower bound for the second formulation of the bridge index and hyperbolic volume of some classes of knots.

This study centers on the concept of S-asymptotically Bloch-type periodicity. We begin by introducing a novel definition of S-asymptotically Bloch-type periodic functions, which generalizes the existing notions of S-asymptotically ω-periodic and S-asymptotically ω-anti-periodic processes. Next, we establish a set of fundamental properties characterizing S-asymptotically Bloch-type periodic processes. Leveraging these results, we then investigate the existence and uniqueness of S-asymptotically Bloch-type periodic mild solutions for certain stochastic semilinear differential equations in Banach spaces. Finally, we explore the global exponential stability of these solutions and present an illustrative example to demonstrate the applicability of our findings.

A new class of fourth-order iterative methods is developed for solving nonlinear equations by using the weight functions. Based on the optimality principles of the Kung-Traub conjecture, the proposed method is optimal while maintaining computational efficiency. Furthermore, the effectiveness of these methods has been demonstrated through testing with nonlinear functions and comparison with established schemes, showcasing the favorable qualities of this new approach. Furthermore, the qualification criteria further highlight the effectiveness of the proposed method.

This paper presents a novel chaotic system based on a modified Chua's circuit model, incorporating a distinctive two-variable modulus function g(x, y) and exhibiting a 2-torus behavior. The system's dynamical properties are analyzed using bifurcation diagram, Poincaré sections, Lyapunov exponents, and phase portraits. A comprehensive stability analysis is conducted by deriving and examining the generalized equilibrium points as functions of system parameters. Additionally, the renowned combination-combination concept, based on a two-driveresponse configuration technique, is integrated, analyzed, and examined using sine, cosine, and exponential functions. The synchronization outcomes are then verified through Lyapunov stability. Numerical simulations are performed to compare theoretical and practical results. Finally, an electronic circuit is designed for the system, with experimental validation provided through oscilloscope measurements.