and Applied Analysis 3 With respect to the pointwise order u ≤ v ⇐⇒ ( ∀x ∈ Ω : u (x) ≤ v (x) ) (11) the setNL(Ω) is a Dedekind complete lattice. In particular, the supremum and infimum of a setA ⊂ NL(Ω) is given by inf A = (I ∘ S) (φ) , (12) supA = (I ∘ S) (ψ) , (13) respectively, where φ : Ω ∋ x 󳨃→ inf{u(x) : u ∈ A} and ψ : Ω ∋ x 󳨃→ sup{u(x) : u ∈ A}. Furthermore, the lattice NL(Ω) is fully di...

Suppose that Φ is a reduced irreducible root system, R is an associative commutative ring with unity, G(Φ, R) is the corresponding adjoint Chevalley group, and E(Φ, R) is its elementary subgroup (see Section 5). There are a lot of results (see, e.g., [Wat80], [Pet82], [GMi83], [HO’M89], [Abe93], [Che00], [Bun07], and references therein*) asserting that, under some conditions, all automorphisms ...

It is well known that given a poset, X, the lattice of order ideals of X, 〈I(X),⊆〉, is a completion of X via the order-embedding φ : X ↪→ I(X) where φ(x) = ↓x. Herein we define a lattice of antichains in X, 〈A(X),4〉, and prove it is isomorphic to 〈I(X),⊆〉. We establish the “join” and “meet” operations of the lattice, and present results for 〈A(X),4〉 analogous to standard results for 〈I(X),⊆〉, i...

Let $k$ be a global field, let $A$ Dedekind domain with $\text{Quot}(A) = k$, and $K$ finitely generated field. Using unified approach for both elliptic curves Drinfeld modules $M$ defined over having trivial endomorphism ring, $k= \mathbb{Q}$, $A \mathbb{Z}$ in the former case function its ring of functions regular away from fixed prime latter case, any nonzero ideal $\mathfrak{a} \lhd A$ we p...

Abstract. Let R be a Dedekind scheme, η its generic point, X and V del Pezzo surfaces of degree 1 over R that are Gorenstein Mori fiber spaces (as 3-folds germs over the ground field). We study birational maps φ : X 99K V over R which are isomorphisms over the generic point of R. We put down normal forms of such transformations (in suitable coordinates) and give some properties of X and V . In ...

Let V (n) denote the number of positive regular integers (mod n) that are ≤ n, and let Vk(n) be a multidimensional generalization of the arithmetic function V (n). We find the Dirichlet series of Vk(n) and give the extremal orders of some totients involving arithmetic functions which generalize the sum-of-divisors function and the Dedekind function. We also give the extremal orders of other tot...

Let R be a domain and K its quotient-field. For a subset S of K , let FR(S) be the set of polynomials f ∈ K[x] with f(S) ⊆ R and define the R -closure of S as the set of those t ∈ K for which f(t) ∈ R for all f ∈ FR(S). The concept of R -closure was introduced by McQuillan (J. Number Theory 39 (1991), 245–250), who gave a description in terms of closure in P -adic topology, when R is a Dedekind...

The investigation of synthetic jet vortex rings flowing through a honeycomb is measured by two-dimensional particle-image velocimetry. established using free jets generated piston–cylinder apparatus upstream the honeycomb. On that basis, turbulence obtained at circulation Reynolds numbers ReΓ = 3437, 5155, 6874, and 8592, Resj 561, 841, 1121, 1402. effects diameter number (ReM 41–449) length-to...