Proposition 0.1. Let R be a commutative ring. TFAE: 1. Every ideal in R is finitely generated. 2. Every chain of ideals I1 ⊆ I2 ⊆ · · · is stable. That is, there exists n such that In = In+1 = · · · . 3. Every nonempty set of ideals has a maximal element wrt inclusion. Proof. 1. (1) =⇒ (2). I = ⋃ k Ik ideal in R. I is generated by x1, x2, . . . xm. There exists n such that x1, . . . , xm ∈ In. ...

Let P = Ca × Cb be a poset where Ci is the chain 1 < · · · < i. A chain blocker of P is an inlcusionwise minimal subset B ⊆ P with the property that every maximal chain in P contains at least one element of B. In [1] the chain blockers of P are being expressed in term of the Catalan numbers and k fold convolution of the Catalan numbers. In this paper we give a complete description of the chain ...

Assuming ZFC, we prove that CH holds if and only if there exists a cofinal maximal chain of order type ω1 in the Turing degrees.

We prove that in the presence of a maximal giant graviton state in N = 4 SYM, the states dual to open strings attached to the giant graviton give rise to an PSU(2, 2|4) open spin chain model with integrable boundary conditions in the SO(6) sector of the spin chain to one loop order.

The intracellular action on exocytosis of botulinum A toxin and constituent chains was studied using permeabilized isolated nerve endings from the rat neural lobe. The release of the neuropeptide vasopressin was measured by radioimmunoassay. In the presence of the reducing agent dithiothreitol, the two-chain form of botulinum A toxin inhibited vasopressin release induced by 10 microM free calci...

Eccentric distance sum (EDS), which can predict biological and physical properties, is a topological index based on the eccentricity of a graph. In this paper we characterize the chain hexagonal cactus with the minimal and the maximal eccentric distance sum among all chain hexagonal cacti of length n, respectively. Moreover, we present exact formulas for EDS of two types of hexagonal cacti.

We have investigated the effects of chemical alkylation of microsomal membranes on nascent chain binding and translocation. Assays were conducted using either full-length or truncated preprolactin transcripts in combination with a reconstituted membrane system consisting of proteolyzed rough microsomes and the cytoplasmic domain of the signal recognition particle receptor. Treatment of rough mi...

biharmonic surfaces in euclidean space $mathbb{e}^3$ are firstly studied from a differential geometric point of view by bang-yen chen, who showed that the only biharmonic surfaces are minimal ones. a surface $x : m^2rightarrowmathbb{e}^{3}$ is called biharmonic if $delta^2x=0$, where $delta$ is the laplace operator of $m^2$. we study the $l_k$-biharmonic spacelike hypersurfaces in the $4$-dimen...