Let $D$ be a two-dimensional regular local ring. We prove there is a one-to-one correspondence between closed connected sets in the space of valuation overrings of $D$ that dominate $D$ and the integrally closed local overrings of $D$ that are not essential valuation rings or divisorial valuation rings of $D$.

Let $F$ be a field, let $D$ be a local subring of $F$, and let Val$_F(D)$ be the space of valuation rings of $F$ that dominate $D$. We lift Zariski's connectedness theorem for fibers of a projective morphism to the Zariski-Riemann space of valuation rings of $F$ by proving that a subring $R$ of $F$ dominating $D$ is local, residually algebraic over $D$ and integrally closed in $F$ if and only if there is a closed and connected subspace $Z$ of Val$_F(D)$ such that $R$ is the intersection of the rings in $Z$. Consequently, the intersection of the rings in any closed and connected subset of Val$_F(D)$ is a local ring. In proving this, we also prove a converse to Zariski's connectedness theorem. Our results do not require the rings involved to be Noetherian.

An h-local domain is a domain for which each nonzero prime ideal is contained in a unique maximal ideal and each nonzero element has finite character. In his dissertation, A. Omairi generalizes the notion of an h-local domain to rings with zero-divisors by restricting the definition to regular ideals. In this article, we give a characterization of when C(X) the ring of continuous real-valued functions on a space X is an h-local ring. We are left with more questions than answers.

Recall that a ring is said to be a clean ring if every element can be expressed as the sum of a unit and an idempotent. In one variant of this definition, a ring is said to be a semi-clean ring if every element can be expressed as the sum of a unit and a periodic element. Ye's Theorem [12] states that the group ring Z(p)[C3] is semi-clean, where p is a prime integer and C3 is a cyclic group of order 3. In this article, we generalize Ye's Theorem by demonstrating that, if R is a local ring, then the group ring R[G] is semi-clean if and only if G is a torsion abelian group.

We show that for an infinite sequence {(Rn,mn)}n≥0 of local normalized quadratic transforms of analytically unramified one-fibered Noetherian local domains, the corresponding sequence of Rees valuations converges. This extends a result of [4], where the authors show that for an infinite sequence of local quadratic transforms of regular local rings, the order valuations converge.

We consider infinite sequences {R_n} of successive local quadratic transforms of a regular local ring. Let S denote the directed union of the sequence of regular local rings R_n. We previously showed the existence of a unique limit point V of the family of order valuation rings of the sequence. In this paper, we examine asymptotic properties of this family of order valuations. We link this asymptotic behavior to ring-theoretic properties of S, namely whether S is archimedean and whether S is completely integrally closed. We construct examples of such S that are archimedean and completely integrally closed but not valuation domains. We give an explicit description of V, where the description depends on whether S is archimedean or non-archimedean.

Let { R_n, 𝔪_n }_n ≥ 0 be an infinite sequence of regular local rings with R_n+1 birationally dominating R_n and 𝔪_nR_n+1 a principal ideal of R_n+1 for each n. We examine properties of the integrally closed local domain S = ⋃_n ≥ 0R_n.

We consider the directed union S of an infinite sequence {(R_n, m_n)} of successive local quadratic transforms of a regular local ring (R, m). If dim R = 2, Abhyankar proves that S is a valuation ring. If dim R > 2, Shannon gives necessary and sufficient conditions for S to be a rank 1 valuation domain and Granja gives necessary and sufficient conditions that S be a rank 2 rational rank 2 valuation domain. Granja observes that these are the only cases where S is a valuation domain. If the sequence is along a rank 1 valuation ring V with valuation v, Granja, Martinez, and Rodriguez show that if the infinite sum of the values v(m_n) diverges, then S = V. We prove that this infinite sum is finite if V has rational rank at least 2. We present an example of a sequence whose union S is a rank 2 valuation domain, but whose value group is not Z^2. We also consider sequences of monomial local quadratic transforms and give necessary and sufficient conditions that the union be a rank 1 valuation domain. If it is, it has rational rank d. We string together finite sequences of monomial local quadratic transforms to construct examples where S is a rank 1 valuation domain with rational rank < d.

Let (R,m) be a regular local ring of dimension at least 2. Associated to each valuation domain birationally dominating R, there exists a unique sequence {Rn} of local quadratic transforms of R along this valuation domain. We consider the situation where the sequence {Rn}n≥0 is infinite, and examine ideal-theoretic properties of the integrally closed local domain S=⋃n≥0Rn. Among the set of valuation overrings of R, there exists a unique limit point V for the sequence of order valuation rings of the Rn. We prove the existence of a unique minimal proper Noetherian overring T of S, and establish the decomposition S=T∩V. If S is archimedean, then the complete integral closure S⁎ of S has the form S⁎=W∩T, where W is the rank 1 valuation overring of V.

Let I be a finitely supported complete m-primary ideal of a regular local ring (R, m). We consider singularities of the normalization of the blow-up Proj R[It] of I. A theorem of Lipman implies that the ideal I has a unique factorization as a star-product of special star-simple complete ideals with possibly negative exponents for some of the factors. If the normalization of the projective model Proj R[It] is regular, we prove that it is the regular model obtained by blowing up the finite set of base points of I. Extending work of Lipman and Huneke-Sally in dimension 2, we prove that every local ring S on the normalization of Proj R[It] that is a unique factorization domain is regular. Moreover, if dim S is at least 2 and S dominates R, then S is an infinitely near point to R, that is, S is obtained from R by a finite sequence of local quadratic transforms.

Let (R,m) be a d-dimensional regular local ring with d ≥ 2. The morphism φ : ProjR[mt] → SpecR defines the blow-up of the maximal ideal m of R. Let (R1,m1) be the local ring of a point in the fiber of m defined by φ. Then R1 is a regular local ring of dimension at most d that birationally dominates R. The ring R1 is said to be a local quadratic transform of R. Local quadratic transforms have historically played an important role in algebraic geometry, and in particular in the understanding of regular local rings. Classically, Zariski’s unique factorization theorem for ideals in a 2-dimensional regular local ring in [9] relies on local quadratic transforms in a fundamental way. More recently, Lipman uses similar methods in [7] to prove a unique factorization theorem for a special class of ideals in arbitrary regular local rings. By taking rings of dimension at least 2, iterating the process of local quadratic transforms yields an infinite sequence {(Rn,mn)}n≥0. We consider the directed union of this infinite sequence of local quadratic transforms, and set S = ∪ n≥0 Rn. Since the rings Rn are local rings that are linearly ordered under domination, S is local, and since the rings Rn are integrally closed, so is S. However, if S is not a discrete valuation ring, then S is not Noetherian. On the other hand, one may consider a valuation ring (V,mV ) that dominates R. There is a unique local quadratic transform R1 of R that is dominated by V , called the local quadratic transform of R along V . If V is the order valuation ring of R, then R1 = V , but otherwise, one may take the local quadratic transform R2 of R1 along V . Iterating this process yields a possibly infinite sequence {(Rn,mn)} of local quadratic transforms, where this process terminates if V is the order valuation ring of some Rn. Abhyankar proves in [1, Proposition 4] that this sequence is finite if and only if the transcendence degree of V/mV over R/m is d−1 (that is, by the dimension formula, the residual transcendence degree is as large as possible). Otherwise, the induced sequence is infinite, and we call the directed union S the Shannon extension of R along V . Given an infinite sequence {(Rn,mn)}n≥0 of local quadratic transforms whose union is S, a basic result in commutative algebra implies there exists some valuation ring V that dominates S. Then a posteriori, the sequence arises by taking local quadratic transforms along V . It is a natural question to ask for conditions in order that a Shannon extension S is a valuation ring. This question has been extensively studied in [1], [8], [3], [2], and [4]. Abhyankar shows in [1, Lemma 12] that if dimR = 2, it is always the case that S is a valuation ring. However, if dimR ≥ 3, then S is sometimes but not always a valuation domain [8]. Conditions in order that a Shannon extension be a valuation ring have been obtained that involve the essential prime divisors of the regular local rings Rn. Shannon shows in [8, Proposition 4.18] that if there is no such essential prime divisor that contains S, then S is a rank 1 valuation ring. Granja shows in [3,

Let I be a finitely supported complete m-primary ideal of a regular local ring (R, m). A theorem of Lipman implies that I has a unique factorization as a *-product of special *-simple complete ideals with possibly negative exponents for some of the factors. The existence of negative exponents occurs if the dimension of R is at least 3 because of the existence of finitely supported *-simple ideals that are not special. We consider properties of special *-simple complete ideals such as their Rees valuations and point basis. Let (R, m) be a d-dimensional equicharacterstic regular local ring with m = (x_1, ..., x_d)R. We define monomial quadratic transforms of R and consider transforms and inverse transforms of monomial ideals. For a large class of monomial ideals I that includes complete inverse transforms, we prove that the minimal number of generators of I is completely determined by the order of I. We give necessary and sufficient conditions for the complete inverse transform of a *-product of monomial ideals to be the *-product of the complete inverse transforms of the factors. This yields examples of finitely supported *-simple monomial ideals that are not special. We prove that a finitely supported *-simple monomial ideal with linearly ordered base points is special *-simple.

This work studies a mathematical model for the dynamics of Chagas disease, a parasitic disease that affects humans and domestic mammals throughout rural areas in Central and South America. It presents a modified version of the model found in Spagnuolo et al. [A model for Chagas disease with controlled spraying, J. Biol. Dyn. 5 (2011), pp. 299-317] with a delayed logistic growth term, which captures an overshoot, beyond the vector carrying capacity, in the total vector population when the blood meal supply is large. It studies the steady states of the system in the case of constant coefficients without spraying, and the analysis shows that for given-averaged parameters, the endemic equilibrium is stable and attracting. The numerical simulations of the model dynamics with time-dependent coefficients are shown when interruptions in the annual insecticide spraying cycles are taken into account. Simulations show that when there are spraying schedule interruptions, spraying may become ineffective when the blood meal supply is large.