One of the greatest difficulties encountered by all in their first proof intensive class is subtly assuming an unproven fact in a proof. The purpose of this note is to describe a specific instance where this can occur, namely in results related to unique factorization and the concept of the greatest common divisor. The Fundamental Theorem of Arithmetic states that every integer exceeding 1 can ...

I have one update from last time, and this is aimed more at the experts. Rob pointed out that there was no reason that we know that a Cartier divisor can be expressed as a difference (or quotient) of effective Cartier divisors. More precisely, a Cartier divisor can be described cohomologically as follows. Let X be a scheme. We have a sheaf O of invertible functions. There is another sheaf K tha...

We describe the big cone of a projective symmetric variety. Moreover, we give a necessary and sufficient combinatorial criterion for the bigness of a nef divisor (linearly equivalent to a G-stable divisor) on a projective symmetric variety. When the variety is toroidal, such criterion has an explicitly geometrical interpretation. Finally, we describe the spherical closure of a symmetric subgrou...

A divisor cordial labeling of a graph G with vertex set V is a bijection f from V to {1, 2, 3, . . .,|V|} such that if an edge uv is assigned the label 1 if f(u) divides f(v) or f(v) divides f(u) and 0 otherwise, then the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. If a graph has a divisor cordial labeling, then it is called divisor cordial graph. ...

In 1952 H.E. Richert by means of the theory of Exponents Pairs (developed by J.G. van der Korput and E. Phillips ) improved the above O-term ( see [8] or [4] pag. 221 ). In 1969 E. Krätzel studied the three-dimensional problem. Besides, M.Vogts (1981) and A. Ivić (1981) got some interesting results which generalize the work of P.G. Schmidt of 1968. In 1987 A.Ivić obtained Ω-results for ∫ T 1 ∆ ...

where the function f(n) is n, n or n + n + p+1 4 , where p ≡ 3 mod 4 is a rational prime, and where dα(n) = #{d : d|n and 1 ≤ d ≤ α} for real α ≥ 1. Motivation for considering these sums comes from an expression which is derived for the class number of a quadratic field with discriminant −p, in terms of a certain restricted divisor sum. This sum is currently too difficult to estimate, in that t...