Let k be a number field. A natural question is: Does k admit an infinite unramified extension? The answer is no, if the root discriminant of k is less than Odlyzko’s bounds. The answer is yes, if k fails the test of Golod-Shafarevic for a prime number p. In that case, we know that there exists an infinite unramified p-extension L over k. But generally it is fairly difficult to determin whether ...

Let $$S=K[x_1,\ldots ,x_n]$$ , where K is a field, and $$t_i$$ denotes the maximal shift in minimal graded free S-resolution of algebra S/I at degree i. In this paper, we prove:

The k-packing problem asks for a subset S of the nodes in a graph such that the distance between any pair of nodes in S is greater than k. This problem has applications to placing facilities in a network. In the current paper we present a self-stabilizing algorithm for computing a maximal k-packing in a general graph. Our algorithm uses a constant number of variables per node. This improves the...

Let G be a simple graph on n vertices having edge-connectivity 1(' (G) > O. We say G is k-critical if 1(' (G) :::: k and 1(1 (G-e) < k for every edge e of G. We denote by ~(n,k) the set of all k-critical graphs on n vertices. In this paper we prove that the maximum number of edges of a graph G in ~(n,k) to be: k(n-k) if n ~ 3k; and L ~ (n+k)2 J. if k + 1 s n < 3k. Further, we characterise the e...

A musical scale can be viewed as a subset of notes or pitches taken from a chromatic universe. Given integers (N,K) N > K we use particular integer partitions of N into K parts to construct distinguished sets, or scales. We show that a natural geometric realization of these sets have maximal area, so we call them maximal area sets. We then discuss properties of maximal area sets for the integer...

, I≤k (n, t) = I (n, t) ∩ 2 ( [n] ≤k ) . After the maximal families in I (n, t) [13] and in Ik (n, t) [1, 9] are known we study now maximal families in I≤k (n, t). We present a conjecture about the maximal cardinalities and prove it in several cases. More generally cardinalities are replaced by weights and asymptotic estimates are given. Analogous investigations are made for I (n, t) ∩ C(n, s),...

given a non-abelian finite group $g$, let $pi(g)$ denote the set of prime divisors of the order of $g$ and denote by $z(g)$ the center of $g$. thetextit{ prime graph} of $g$ is the graph with vertex set $pi(g)$ where two distinct primes $p$ and $q$ are joined by an edge if and only if $g$ contains an element of order $pq$ and the textit{non-commuting graph} of $g$ is the graph with the vertex s...

1 Preliminary definitions Definition 1 (S-Lenstra constant) Let K be a number field, and Z K its ring of integers. Let S be a finite set of valuations of K containing all the infinite valuations. Let L be a maximal subset of Z K such that x, y ∈ L ⇒ x − y ∈ Z if L = {a 0 , a 1 ,. .. , a k }. In fact we will use this second form of the condition. Remark 2 We can assume that if L is such a maxima...

Brain capillary endothelial cells play an important role in ion homeostasis of the brain through the transendothelial transport of Na and K. Since little is known about the regulation of ion transport in these cells, we determined the effect of extracellular potassium concentration ([K]o) on the kinetics of the Na,K-pump in isolated cerebral microvessels using both K uptake and Na efflux as mea...

the aim of the present study was to examine the effect of aerobic training and green tea supplementation on serum leptin and insulin resistance in overweight and obese men. for this purpose, 43 overweight and obese men were selected voluntarily and were included in four groups: control (body mass index 31.4+3.7, vo2max 39.3+3.8 ml/kg.min), green tea supplement (body mass index, 30.3±2.8 k/m2, v...