In this paper we investigate 1/2 -derivations, local 1/2 -derivations, and 2-local 1/2 -derivations of Lie algebras related to model nilpotent Lie algebras. We first obtain an explicit description of the general form of 1/2 -derivations on model algebras. We also demonstrate that model algebras admit local and 2-local 1/2 -derivations which are not 1/2 -derivations. Further, we study solvable Lie algebras whose nilradical is a model Lie algebra, derive the general form of 1/2 -derivations for these solvable extensions, and prove a rigidity result: for solvable Lie algebras with model nilradical and maximal dimension of the complementary space, all local and 2-local 1/2 -derivations coincide with 1/2 -derivations.
We consider the following elliptic system -Δ u_i=(|x|^-μ*|u_i|^4-μ/2)|u_i|^2-μ/2u_i+∑ _j=1 j≠i^2β u_iu_j^2 in ℝ^4, i=1,2. If β <0, |β | is small enough we build solutions such that the first component looks like the radial positive solution of the single equation, while the second one blows-up at the k vertices of a regular polygon.
Rubio de Francía’s extrapolation results are presented in one-sided setting, generally speaking, in grand ball Banach function spaces. These results can be applied, for example, to obtain the boundedness of one-sided operators of Harmonic Analysis for those operators for which one-sided Muckenhoupt condition guarantees the one–weight inequality.
In this paper, we investigate the regularizing effect arising from the interplay between coefficients in quasilinear problems involving X-elliptic operators, a class of degenerate operators introduced by Lanconelli and Kogoj in [15]. Due to the presence of a natural growth term, the problem lacks a variational structure. By combining a suitable approximation scheme with Schauder’s fixed point theorem, we prove the existence of bounded weak solutions if the coefficients satisfy the Q-condition introduced by Arcoya and Boccardo in [1].
The study proposed in this paper is part of a larger scientific collaboration involving a large academic community, and it is specifically focused on the modeling aspects of advection–diffusion problems in water monitoring and distribution systems. The main idea proposed in this work is the study of an anomalous advection–diffusion process for water distribution systems. Indeed, the dynamics of water within distribution systems is characterized by different velocity regimes, ranging from laminar, transitional and turbulent, which lead to an advective and diffusive-dispersive behavior of the contaminant spread. This phenomenon, usually modeled by a classical advection-dispersion-reaction equation, through the introduction of a fractional derivative operator, will be modeled by a new fractional model, of advection-dispersion-reaction equation, which allows to describe anomalous processes. The fractional model has been studied numerically, attempting the fitting real phenomena.
We study the global boundedness of weak solutions to a class of nonlinear elliptic equations in bounded Lipschitz domains, subject to conormal boundary conditions. Using techniques from Morrey space theory, Adams-Maz’ya trace inequalities, and higher integrability properties of the gradient, we derive uniform a priori bounds for the solutions. A crucial step in our analysis is the construction of appropriate measures associated with the solution and the application of the Hartman–Stampacchia lemma, which enables us to control the solution up to the boundary of the domain.
Abstract This paper introduces and investigates new generalizations of both 1-absorbing prime and $$\rho $$ ρ -ideals in the setting of noncommutative rings with identity, using special radicals $$\rho $$ ρ . We define the notions of $$(1,\rho )$$ ( 1 , ρ ) -absorbing and $$(1,\rho ^*)$$ ( 1 , ρ ∗ ) -absorbing ideals for a given special radical $$\rho $$ ρ , and we study the relationships between these new classes of ideals and previously known ideal classes. Several structural results are obtained, characterizing when these ideals coincide or differ, especially in the existence or absence of the identity. We provide characterizations in product rings, under homomorphic images, and for constructions such as idealizations. A portion of the study is devoted to the case when $$\rho = \mathcal {P}$$ ρ = P , the prime radical, where we establish specific properties in various classes of rings, including CI -rings.
Let G be a totally disconnected locally compact (tdlc) group. We denote by ( G) the space of closed subgroups of G equipped with the Chabauty topology. A closed subgroup H of G is called locally elliptic if every compact subset of H is contained in a compact subgroup. In this paper, we address the following quention: Is the collection _ℒℰ( G) of closed locally elliptic subgroups of G a closed subset of the Chabauty space ( G) ? We prove that the space of _ℒℰ( G) is Chabauty-closed when the group G contains an open solvable subgroup.
In this note, we revisit recent Liouville-type and rigidity results for the CR Yamabe equation established in [4, 5]. By refining their arguments, we obtain improved Liouville theorems and rigidity statements on complete noncompact Sasakian manifolds which is stated in dimension 5 in [5]. Moreover, we provides a short and self-contained proof of a Liouville theorem given in [4] on the Heisenberg group. Our approach combines integral identities of Jerison–Lee type with suitable energy estimates. A novel thing is that we do not need to use the test function as in [5].
This article introduces the concept of geodesic ratio invex functions, extending the framework of invexity to the geodesic setting. Within this context, necessary and sufficient conditions are established for the existence of optimal solutions to multiobjective fractional programming problems. Furthermore, various duality theorems–including weak, strong, and strict converse duality–are developed under geodesic ratio invexity assumptions. Illustrative examples are provided to demonstrate the applicability of the proposed conditions and to validate the established duality relationships.
The main purpose of this short note is to extend the Banach contraction principle to a relatively broad and easily describable class of topological vector spaces, namely sequentially complete locally convex topological vector spaces. To this end, we introduce a notion of contraction in topological vector spaces and prove a fixed point theorem for such mappings when the underlying space is sequentially complete and locally convex. We also provide several examples and remarks.
A subgroup H of a finite group G is said to be an ℋ -subgroup of G if N_G(H)∩ H^g≤ H for all g∈ G ; and H is said to be a weakly ℋ -embedded in G, if there exists a normal subgroup K of G such that H^G=HK and N_G(H∩ K)∩ (H∩ K)^g≤ H∩ K , for all g∈ G . In this paper, we continue the research on weakly ℋ -embedded subgroups introduced by Asaad and Ramadan (2016) [On weakly ℋ -embedded subgroups of finite groups. Commun. Algebra 44:4564-4574].
The maximum principle for elliptic systems can be derived either from analytic properties of the system or from features of the associated variational functional. In this paper, we compare a componentwise sign condition guaranteeing the maximum principle in the PDE formulation with a monotonicity condition employed in the variational framework.
The aim of this paper is to derive some Calderón-Zygmund estimates for local solutions u of nonlinear elliptic systems of the type divA(x, Du) = div|G|^p - 2G in Ω⊂ℝ^n, where Ω⊂ℝ^n is a bounded domain. We assume that u: Ω→ℝ^N belongs to a weighted Sobolev space W_loc^1, p , with p ∈( 2n/n + 2, 2) , G belongs to a weighted L_loc^p space and x →A(x, ξ ) has growth coefficients in the John and Nirenberg space BMO. As an application of similar techniques, we also consider weak solutions u of linear elliptic systems of the type div(A(x)Du) = div G, where the matrix A(x) lies in the space BMO. In this case, an improvement of the admissible exponent range in the Calderón-Zygmund estimate is obtained.
Let ℛ be a nonzero prime ring with characteristic other than 2 and ℒ be a non central Lie ideal of ring ℛ. Assume that 𝒯_1 and 𝒯_2 are b-generalized skew derivations on ℛ such that p (𝒯_1(x)x-x𝒯_2(x) )=0 for all x ∈ℒ. We examine complete possible structures of 𝒯_1 and 𝒯_2. This extends the result of Rania [19. Communications in Algebra]
Let S be a semigroup, and D a finite directed graph with a nonempty edge set. The directed power graph of S is the directed graph which has all elements of S as vertices and has edges (x, y) for all x, y∈S such that x y and y is a power of x. We say that S is power D -saturated if for every infinite subset T of S, the directed power graph of S contains a subgraph isomorphic to D with all vertices in T. Let Δ be a group with identity ε , and let {S_δ}_δ∈Δ be a family of subsets of a semigroup S with zero 0 such that: i) S=⋃ _δ∈ΔS_δ ; ii) S_δ∩S_γ ={0} for all distinct δ , γ∈Δ ; iii) S_δS_γ =S_δγ for all δ , γ∈Δ . Then we say that S is a strong homogeneous semigroup, and graded by Δ , or that S is a strongly Δ -graded semigroup. A semigroup S without zero is said to be strongly Δ -graded if S^0=S∪{0} is. We characterize power D -saturation of an infinite strongly Δ -graded semigroup S in terms of power D -saturation of S_ε . In particular, for an infinite group G, we prove that G is power D -saturated if and only if the center C(G) of G is not simple, both N and G/N are power D-saturated for every subgroup N of G, contained in C(G), and the index of C(G) is not divisible by p for each quasicyclic p-subgroup of G .
Starting from a mixed Boltzmann–BGK model for binary gas mixtures, we formally derive Navier–Stokes equations in different regimes, in the asymptotic limit for proper Knudsen number, with explicit computation of the transport coefficients of viscosity and thermal conductivity. First, we focus on the regime dominated by the whole collision phenomena; then, we consider the case of ε –mixtures of heavy and light species. In this latter case we assume, on one hand, a two–scale collision regime, with interactions within each component that constitute the dominant process, and, on the other hand, a three–scale regime, with dominant collisions within the heavier species and the slowest interactions occurring within the lighter one, leading to fluid–kinetic equations.
We investigate a variational problem for eigenvalues of the Laplace-Beltrami operator on smooth manifolds with respect to Radon measures belonging to a suitable class; we are motivated by conformal eigenvalues in dimension two. Our main result is a regularity result for stationary measures with respect to outer variations. More precisely, we prove that any sufficiently regular stationary measure is absolutely continuous with respect to the classical volume measure and that its density is induced by a harmonic map. Our result has some interesting applications to Steklov eigenvalues on subdomains.
