In this work, we study idempotent preradicals with the properties of being hereditary, radical, and stable. We prove that if every idempotent preradical is radical, then every semiartinian left R-module is semisimple. Also, if every idempotent preradical is hereditary, then R is a left self-injective and a left MAX ring. And, if every idempotent preradical is stable, then R is a left V-ring and a left Noetherian ring. Finally, we characterize the rings where every idempotent preradical is a hereditary radical; in the commutative case R must be semisimple, and in the noncommutative case R must be semisimple and must also satisfy that the left annihilator of every element in R is a two sided ideal of R.
We determine sharp radii of Ma–Minda starlikeness for the class 𝒲_n(β ) consisting of functions f∈𝒜_n satisfying Re (f(z)/z)>β , where 0≤β <1 . A sharp estimate for the logarithmic derivative zf'(z)/f(z) is first obtained, which leads to a unified radius theorem. By combining this estimate with known disk inclusion results for several Ma–Minda domains, we obtain sharp radius constants for the classes associated with the nephroid, sine, lune, sigmoid, and hyperbolic starlike functions. The sharpness of each radius is established by exhibiting an extremal function.
Product formulas for γ , e^γ and rational approximations for γ are given. This approximation is based on the construction of a linear form in logarithms.
An outer multiset resolving set uniquely identifies the vertices of a given graph. A subset W ⊆ V(G) is an outer multiset resolving set if every two distinct vertices in V(G)∖ W have distinct multisets of distances to W. The smallest such set is an outer multiset basis, and its cardinality is the outer multiset dimension of G. This notion is closely related to the multiset resolving set, which distinguishes all vertices of G, while outer multisets only distinguish those outside the set. In this paper, for integers d,c ≥ 1 , we construct an infinite family of graphs with diameter d admitting a multiset resolving set W = {w_1, … , w_c} of maximum order n = d + c - 1 ()d - 1 + c . Besides, we are interested in finding the outer multiset dimension of an infinite family of graphs. Since the outer multiset dimension is known for cycles, we took the circulant graphs C_n(1,2) of order n as a natural generalization of the cycles.
This paper provides a rigorous and self-contained proof of the Jacobson lemma for the Jeribi essential spectrum. Under a spectral compatibility assumption (Hypothesis 2.1), namely that the Jeribi essential spectrum of an operator coincides with the spectrum of its coset in the Calkin quotient algebra ℒ(Z)/𝒲(Z) for Z=X,Y , we establish that the Jeribi essential spectrum is invariant under permutation of operators: σ _j(BC)∖{0} = σ _j(CB)∖{0}. The proof is entirely algebraic and takes place in the Calkin quotient algebra; no lifting of invertibility from the quotient to the original algebra is required. Under the same assumption, we extend our results to block operator matrices, giving characterizations of the Jeribi essential spectrum for even and odd powers of off-diagonal operators. The analysis is accompanied by examples and counterexamples that illustrate the role of the hypothesis.
