the aim of this paper is to study orders over a valuation ring v with arbitrary rank in acentral simple f-algebra q. the relation between all of the orders is explained with a diagram. it is thenshown that inside bezout order, extremal v-orders are precisely semi-hereditary. in the last section, theeffect of henselization on maximal and semi-hereditary orders is examined.

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 V (n) denote the number of positive regular integers (mod n) less than or equal to n. We give extremal orders of V (n)σ(n) n2 , V (n)ψ(n) n2 , σ(n) V (n) , ψ(n) V (n) , where σ(n), ψ(n) are the sum-of-divisors function and the Dedekind function, respectively. We also give extremal orders for σ∗(n) V (n) and φ∗(n) V (n) , where σ∗(n) and φ∗(n) represent the sum of the unitary divisors of n a...

Let G be a connected k–regular bipartite graph with bipartition V (G) = X ∪Y and adjacency matrix A. We say G is det–extremal if per(A) = |det(A)|. Det–extremal k–regular bipartite graphs exist only for k = 2 or 3. McCuaig has characterized the det–extremal 3–connected cubic bipartite graphs. We extend McCuaig’s result by determining the structure of det–extremal cubic bipartite graphs of conne...

Let A be a subset of {1, 2, . . . , n} such that the sum of no two distinct elements of A is a prime number. Such a subset is called a prime-sumset-free subset of {1, 2, . . . , n}. A prime-sumset-free subset is called an extremal prime-sumset-free subset of {1, 2, . . . , n} if A ∪ {a} is not a prime-sumset-free subset for any a ∈ {1, 2, . . . , n} \ A. We prove that if n ≥ 10 then there is no...

Interval orders and interval graphs are particularly natural examples of two widely studied classes of discrete structures: partially ordered sets and undirected graphs. So it is not surprising that researchers in such diverse fields as mathematics, computer science, engineering and the social sciences have investigated structural, algorithmic, enumerative, combinatorial, extremal and even expe...

The present paper aims to point out how the stationary-excess operator and its iterates transform the s-convex stochastic orders and the associated moment spaces. This allows us to propose a new unified method on constructing s-convex extrema for distributions that are known to be t-monotone. Both discrete and continuous cases are investigated. Several extremal distributions under monotonicity ...

We study the exact extremal orders of compositions f(g(n)) of certain arithmetical functions, including the functions σ(n), φ(n), σ∗(n) and φ∗(n), representing the sum of divisors of n, Euler’s function and their unitary analogues, respectively. Our results complete, generalize and refine known results. Mathematics Subject Classification: 11A25, 11N37