In this paper we study rank one discrete valuations of the field k((X1, . . . ,Xn)) whose center in k[X1, . . . ,Xn] is the maximal ideal. In sections 2 to 6 we give a construction of a system of parametric equations describing such valuations. This amounts to finding a parameter and a field of coefficients. We devote section 2 to finding an element of value 1, that is, a parameter. The field o...

In this article, we introduce the fractional Hardy–Littlewood maximal function on infinite rooted k-ary tree and study its weighted boundedness. We also provide examples of weights for which satisfies strong type (p, q) estimates tree.

Let k be a number field, suppose that B is a central simple division algebra over k, and choose any maximal order D of B. The object of this paper is to show that the group DS of S-units of B is generated by elements of small height once S contains an explicit finite set of places of k. This generalizes a theorem of H. W. Lenstra, Jr., who proved such a result when B = k. Our height bound is an...

We investigated maximal Prym varieties on finite fields by attaining their upper bounds on the number of rational points. This concept gave us a motivation for defining a generalized definition of maximal curves i.e. maximal morphisms. By MAGMA, we give some non-trivial examples of maximal morphisms that results in non-trivial examples of maximal Prym varieties.

Let E/Q be an elliptic curve defined over Q of conductor N and let Gal(Q/Q) be the absolute Galois group of an algebraic closure Q of Q. For an automorphism σ ∈ Gal(Q/Q), we let Q be the fixed subfield of Q under σ. We prove that for every σ ∈ Gal(Q/Q), the Mordell-Weil group of E over the maximal Galois extension of Q contained in Q σ has infinite rank, so the rank of E(Q σ ) is infinite. Our ...

Substitute valuations (in some contexts called gross substitute valuations) are prominent in combinatorial auction theory. An algorithm is given in this paper for generating a substitute valuation through Monte Carlo simulation. In addition, the geometry of the set of all substitute valuations for a fixed number of goods K is investigated. The set consists of a union of polyhedrons, and the max...

An edge cut that separates the connected graph into components with order at least two is restricted edge cut. The cardinality of minimum restricted edge cut is restricted edge connectivity. Denote byλ′ ( G) the restricted edge connectivity , then λ′( G) ≤ξ( G) where ξ( G) is the minimum edge degree. G is called maximal restricted edge connected if the equality in the previous inequality holds....

Let G = (V, E) be an undirected graph without loops and multiple edges. A subset C ⊆ V is called identifying if for every vertex x ∈ V the intersection of C and the closed neighbourhood of x is nonempty, and these intersections are different for different vertices x. Let k be a positive integer. We will consider graphs where every ksubset is identifying. We prove that for every k > 1 the maxima...