Divisor

About: Divisor is a research topic. Over the lifetime, 2462 publications have been published within this topic receiving 21394 citations. The topic is also known as: factor & submultiple.

Inter-node data transfer method and inter-computer data transfer method

Abstract: PROBLEM TO BE SOLVED: To execute data transfer which makes the whole data to be transferred exist on the whole memories in the shortest time. SOLUTION: This data transfer method performs one side transfer and mutual transfer with other computers and is performed by a parallel computer that consists of plural computers provided with a memory and performs the fastest data transfer by performing mutual transfer two times after performing one side transfer two times at the time of transferring data so as to be stored in the whole computers, for instance, when data D(1) is stored is M(12) among computers 1 to 16 (memories M(1) to M(16)), D(2) in M(10), D(3) in M(11) and D(4) in M(12). Generally, when there are M computers (however, M is the power of 2) and D pieces (however, 1

The Rational Number n/p as a sum of two unit fractions

Abstract: In a 2011 paper published in the journal "Asian Journal of Algebra"(see reference[1]), the authors consider, among other equations,the diophantine equations 2xy=n(x+y) and 3xy=n(x+y). For the first equation, with n being an odd positive integer, they give the solution x=(n+1)/2, y=n(n+1)/2. For the second equation they present the particular solution, x=(n+1)/3,y=n(n+1)/3, where is n is a positive integer congruent to 2modulo3. If in the above equations we assume n to be prime, then these two equations become special cases of the diophantine equation, nxy=p(x+y) (1), with p being a prime and n a positive integer greater than or equal to 2. This 2-variable symmetric diophantine equation is the subject matter of this article; with the added condition that the intager n is not divisible by the prime p. Observe that this equation can be written in fraction form: n/p= 1/x + 1/y(See [2] for more details) In this work we prove the following result, Theorem1(stated on page2 of this paper):Let p be a prime, n a positive integer at least2, and not divisible by p. Then, 1)If n=2 and p is an odd prime, equation (1) has exactly three distinct positive integer solutions:x=p, y=p ; x=p(p+1)/2, y=(p+1)/2 ; x=(p+1)/2, y=p(p+1)/2 2)If n is greater than or equal to 3, and n is a divisor of p+1. Then equation (1) has exactly two distinct solutions: x=p(p+1)/n, y=(p+1)/n ; x=(p+1)/n, y=p(p+1)/n 3) if n is not a divisor of p+1. Then equation (1) has no positive integer solution. The proof of this result is elementary, and only uses Euclid's Lemma from number theory,and basic divisor arguments(such that if a prime divides a product of two integers; it must divide at least one of them).

On Hilbert covariants

Abstract: Let F denote a binary form of order d over the complex numbers. If r is a divisor of d, then the Hilbert covariant H_{r,d}(F) vanishes exactly when F is the perfect power of an order r form. In geometric terms, the coefficients of H give defining equations for the image variety X of an embedding P^r->P^d. In this paper we describe a new construction of the Hilbert covariant; and simultaneously situate it into a wider class of covariants called the G\"ottingen covariants, all of which vanish on X. We prove that the ideal generated by the coefficients of H defines X as a scheme. Finally, we exhibit a generalisation of the G\"ottingen covariants to n-ary forms using the classical Clebsch transfer principle.

On Q-conic bundles, II

Abstract: A $\mathbb Q$-conic bundle germ is a proper morphism from a threefold with only terminal singularities to the germ $(Z i o)$ of a normal surface such that fibers are connected and the anti-canonical divisor is relatively ample. We obtain the complete classification of $\mathbb Q$-conic bundle germs when the base surface germ is singular. This is a generalization of our previous paper math/0603736, which further assumed that the fiber over $o$ is irreducible.

Canonical Rings of \({\mathbb{Q}}\)-Divisors on \({\mathbb{P}^1}\)

Abstract: The canonical ring \({S_{D} = \oplus_{d\geq0}H^{0}(X, \lfloor dD\rfloor)}\) of a divisor D on a curve X is a natural object of study; when D is a \({\mathbb{Q}}\)-divisor, it has connections to projective embeddings of stacky curves and rings of modular forms. We study the generators and relations of \({S_D}\) for the simplest curve \({X = \mathbb{P}^1}\). When D contains at most two points, we give a complete description of \({S_D}\); for general D, we give bounds on the generators and relations. We also show that the generators (for at most five points) and a Grobner basis of relations between them (for at most four points) depend only on the coefficients in the divisor D, not its points or the characteristic of the ground field; we conjecture that the minimal system of relations varies in a similar way. Although stated in terms of algebraic geometry, our results are proved by translating to the combinatorics of lattice points in simplices and cones.