Let 𝖆 be an ideal of a Noetherian ring R and M be a finitely generated R-module with cd(𝖆, M) = c ≥ 0. Assume that the R-module H^c_𝖆 (M) is Artinian and dim Hⁱ_𝖆 (M) ≤ 1 for all integers i ≥ 2. In this paper, it is shown that the R-module H^c_𝖆 (M) is 𝖆-cofinite. Also, in this direction some of other results will be included.

It is known that the power series ring R[[x]] over a Dedekind finite ring R is Dedekind finite. Then the polynomial ring R[x] over R is also Dedekind finite as a subring of R[[x]]. But this procedure does not show the useful details that occur in the argument of the direct proof of Dedekind finiteness passing to polynomial rings. Thus, in this article, we provide R[x] with a direct proof, to study the structure of coefficients of the products of polynomials.

We prove the Kollár conjecture for the singularity W_p,q,r, where W_p,q,r is a weighted homogeneous surface singularity that admits a smoothing whose Milnor fiber is a rational homology disk. That is, we prove that every one-parameter smoothing of W_p,q,r is induced by one of its P-resolutions.

This paper introduces and investigates the concept of restricted genus for finitely generated groups with finite commutator subgroups. The restricted genus extends classical notions of genus in group theory by incorporating homomorphism data, providing a refined approach to group classification beyond isomorphisms. For a finitely generated group G with a finite commutator subgroup, a finite group F, and a homomorphism h : F → G, we define the restricted genus γ(G, h) as the collection of group-homomorphism pairs (H, h') where H is π_G-equivalent to G for a specific set of primes π_G, and h' is a homomorphism from F to H that is π_G-equivalent to h. We establish theoretical foundations for this concept, develop an algorithmic framework for computing the restricted genus, and demonstrate its properties through examples including dihedral groups, quaternion groups, and semidirect products. We prove that the restricted genus distinguishes between groups that may appear similar under other equivalence relations, while revealing unexpected connections between structurally different groups. Furthermore, we establish connections between the restricted genus and cohomological invariants, providing new insights into the classification of group extensions. Our work bridges pure algebraic structures with computational methods, offering both theoretical insights and practical tools for understanding the intricate relationships between finitely generated groups with finite commutator subgroups.

Let A = (A_i)_i≥0 be an ascending chain of commutative rings with identity, A[[X]] (respectively A[X]) the ring of power series (respectively polynomial) with coefficient of degree i in A_i for each i ∈ ℕ. We study when the ring A[[X]] (respectively A[X]) satisfies divisibility on ascending chain of ideals.

In this paper, we study a class of generalized Lambert W functions defined by the equation (w² - r)e^w = z, which broadens the scope of transcendental equations that can be solved in mathematics, physics, and engineering. We analyze their basic properties, including real and complex branches, derivatives and integrals, Taylor series expansions, and branch structures. We also develop numerical schemes, such as Halley-type iterations, for higher-order generalizations. Practical applications are presented to illustrate the versatility of these functions, including quantile functions for Erlang and negative binomial distributions, delay differential equations, and physical models such as the double Dirac delta potential.

Let 𝔈 denote a unital *-algebra. For any elements ${\mathfrak{E}},\,{\mathfrak{F}}$ ∈ 𝔈, we define the products ${\mathfrak{E}}{\circledcirc}{\mathfrak{F}}={\mathfrak{E}}^*{\mathfrak{F}}+{\mathfrak{F}}^*{\mathfrak{E}}$ and $[{\mathfrak{E}},{\mathfrak{F}}]_{\bullet}={\mathfrak{E}}^*{\mathfrak{F}}-{\mathfrak{F}}^*{\mathfrak{E}}$, which are referred to as the bi-skew Jordan product and the bi-skew Lie product, respectively. In this article, we establish that a nonlinear mapping ζ : 𝔈 → 𝔈 satisfying $${{\zeta}([{\mathfrak{E}}_1\,{\circledcirc}\,{\mathfrak{E}}_2\,{\circledcirc}\,{\cdots}\,{\circledcirc}\,{\mathfrak{E}}_{n-1},\,{\mathfrak{E}}_n]_{\bullet})\\=\sum_{j=1}^{n}[{\mathfrak{E}}_1\,{\circledcirc}\,{\mathfrak{E}}_2\,{\circledcirc}\,{\cdots}\,{\circledcirc}\,{\mathfrak{E}}_{j-1}\,{\circledcirc}\,{\zeta}({\mathfrak{E}}_j)\,{\circledcirc}\,{\mathfrak{E}}_{j+1}\,{\circledcirc}\,{\cdots}\,{\circledcirc}\,{\mathfrak{E}}_{n-1},\,{\mathfrak{E}}_n]_{\bullet}$$ for all ${\mathfrak{E}}_1,\,{\mathfrak{E}}_2,\,{\cdots},\,{\mathfrak{E}}_n$ ∈ 𝔈 and n ≥ 3, is an additive *-derivation. Furthermore, we apply the above-mentioned result to several distinct algebras.

In this paper, we study the notion of an S-primary-stable ring, that is, a ring in which every S-primary ideal is primary. We present several results including characterizations and the transfer of the S-primary-stable property to homomorphic images and localizations. We also investigate the possible transfer of the S-primary-stable property between a ring A and its constructions A ∝ M and A ⋈_f J. Our results provide new classes of commutative rings satisfying this property.

Let 𝓗 be the set of all commutative rings with unity whose nilradical is a divided prime ideal. For a given ring extension R ⊂ T in class 𝓗 such that Nil(T) = Nil(R), it is shown that if each proper subring of T containing R is 𝜙-integrally closed (resp., 𝜙-PVR), then T is 𝜙-integrally closed (resp., 𝜙-PVR).

Let C^𝔹[a, b] denote an analogue of Wiener space over paths in abstract Wiener space 𝔹, the space of 𝔹-valued continuous functions on [a, b]. In this paper, we introduce a positive finite measure with a scale on C^𝔹[a, b] which is a generalized analogue of Wiener measure. We then investigate its various properties including the Fourier transforms on the space and the translation of the time interval [a, b] defining C^𝔹[a, b]. As applications of the results, we derive various relationships between the analogue of Wiener space and its product space. In particular, we express the measure on C^𝔹[a, b] in terms of a product measure on C^𝔹[a, s]×C^𝔹[s, b] by dividing [a, b] as [a, s] and [s, b]. Using the results, we finally provide various applicable examples which are useful in the theory of a generalized Brownian motion process.

The higher-order Schwarzian derivative Vf proposed by Kim and Sugawa exhibits unique invariance property under composition with Möbius transformations. This paper establishes two principal results: (i) sharp bounds for Vf(0) in subclasses of starlike and convex functions, explicitly relating to geometric parameters α and β; (ii) the holomorphy of higher Bers maps induced by Vf in universal, Weil-Petersson and BMO-Teichmüller spaces, extending Bers' foundational embedding theory.

In this study, a new subclass of analytic and bi-univalent functions is introduced using generalized Laguerre polynomials, which generalizes several existing subclasses. We obtain upper bounds for the initial coefficients and investigate the Fekete-Szegö estimates for functions belonging to the newly defined class. Furthermore, we establish relevant connections between our results and those reported in earlier investigations.

There are several sufficient conditions in the form of differential inequalities for an analytic function to be a function with positive real part in the open unit disk. However, these conditions are not necessary for functions with positive real part. This paper investigates such radius problems by determining the largest subdisk in which these conditions are satisfied. The applications of such radius problems are also highlighted in univalent function theory.

Let 𝖋 be a transcendental entire function with hyper-order less than one. The aim of this note is to study the value distribution of the differential-difference monomials α𝖋(z)^q₀ (𝖋(z + c))^q₁, where c is a non-zero complex number and q₀ ≥ 2, q₁ ≥ 1 are non-negative integers, and α(z) (≢ 0, ∞) is a small function of 𝖋.

In this paper, we study the long-term behavior of the following damped hyperbolic equation ε(t)u_tt + αu_t - ∆_λu + f(u) = g(x), t > τ, where ε(t) is a decreasing function vanishing at infinity. Here, ∆_λ is the strongly degenerate operator and f, g satisfy certain conditions specified later. Using the theory of processes in time-dependent spaces and a variant of the criterion for the asymptotic compactness for associated processes related to our problem, we prove the existence of time-dependent global attractors for the damped hyperbolic equations in time-dependent phase spaces. Our approach is similar to those in [9, 26] and is based on a combination of the Galerkin methods, compactness arguments and the use of standard energy estimates, notably, some of the techniques used here are simpler than those in the referenced works.

In this paper, we introduce the Fredholm, left (right) Fredholm, and Weyl essential pseudospectra of an element in a Banach algebra. We begin by providing their definitions and then explore various properties of these essential pseudospectra, including the stability of the Fredholm, left (right) Fredholm, and Weyl essential pseudospectra. More specifically, our investigation aims to highlight the potential applications of the concept of pseudo Fredholm perturbations in Banach algebras, particularly in relation to subalgebras. This includes analyzing essential pseudospectra with respect to subalgebras in the context of operator matrix fields.

In this paper, we study inclusion relations for a previously introduced class of harmonic close-to-convex functions. Using operators defined via the Pascal and Poisson distribution series and applying suitable coefficient bounds, we obtain new inclusion results among the classes of harmonic convex functions, harmonic starlike functions, and the defined class. Our findings highlight the effectiveness of distribution series operators in geometric function theory.

In this article, we will show some algebraic dependence theorems for meromorphic mappings into a projective space sharing few moving hyperplanes with different multiplicities. Our result is an improvement of many previous results in this topic.

In the context of a complete Riemannian manifold M and a self-adjoint Codazzi (1, 1)-tensor field P defined on it, we establish a Liouville-type theorem concerning the operator ∆_P,V which this operator is generalized of Laplacian and weighted Laplacian. Here, P is a Codazzi self-adjoint (1, 1)-tensor field, and V belongs to the space of vector fields 𝔛(M). We demonstrate that if the inequality $Ric(Y,PY)-Hessf^P(Y,Y)-{\frac{1}{2}}{\mathcal{L}}_Vg(Y,Y){\geq}0$ holds for all Y ∈ 𝔛(M) and if u is a bounded ∆_P,V-harmonic function, then u must be constant. This conclusion is derived from a gradient estimate applicable to ∆_P,V-harmonic functions within smooth metric measure spaces, where the condition $Ric(Y,PY)-Hessf^P(Y,Y)-{\frac{1}{2}}{\mathcal{L}}_Vg(Y,Y){\geq}-(n-1){\delta}_nH^2{\mid}Y{\mid}^2$ is satisfied for all X ∈ 𝔛(M).

Making use of the revenue equivalence principle, this research provides an analysis of a particular N-player war-of-attrition game which concludes when one player resigns. An all-pay auction that is equivalent to the N-player war of attrition of this article and a Vickrey-type auction that bears an identical allocation rule as the all-pay auction of this article are developed. The equilibrium payment of the Vickrey-type auction is readily obtained. Then, using the revenue equivalence principle, the expected payment of the all-pay auction is obtained. Through this approach, the unique symmetric equilibrium of this article's N-player war-of-attrition game is derived. Comparative statics results are also provided.

This paper introduces an alternative approach for two-level overlapping Schwarz methods for solving heterogeneous vector field problems discretized with face finite elements in three dimensions. This approach is developed through the creation of a new type of coarse space, constructed to minimize the energy associated with a subset of the interface terms between subdomains. By focusing on minimizing the energy, the new approach effectively captures the effect of the jumps of coefficients between subdomains. The effectiveness of the proposed method is validated by numerical experiments conducted across various test cases.