Topic
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.
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TL;DR: In this article, it was shown that if S is factor-closed (FC set), then the GCD matrix on S is the greatest common divisor of any two elements of S.
Abstract: Let S = {x1, x2, ..., xn} be a set of distinct positive integer,and let f be an arithmetical function. Then n × n matrix (S) whose i,j-entry is the greatest common divisor (xi, xj) of xi and xj is called the GCD matrix on S.[1, 2, 4, 5] The set S is said to be factor-closed if it contains every divisor of any element of S,and the set S is said to be GCD-closed if it contains the geratest common divisor of any two elements of S.[2, 3, 7] In 1876, H. J. S. Smith showed that if S is factor-closed (FC set), then
2 citations
Posted Content•
TL;DR: A generalization of Kawamata's result in arbitrary dimension was shown in this article. But this result is restricted to Fano manifold of dimension n and index n-3.
Abstract: Let X be a Fano manifold of dimension n and index n-3. Kawamata proved the non vanishing of the global sections of the fundamental divisor in the case n=4. Moreover he proved that if Y is a general element of the fundamental system then Y has at most canonical singularities. We prove a generalization of this result in arbitrary dimension.
2 citations
Patent•
24 Feb 2004
TL;DR: In this article, a divider for dividing a dividend by a divisor is presented, including a subtractor for subtracting the divisors from the dividend to produce a result, storage space with a preliminary answer, and a processor for revising the dividend and preliminary answer based on the result.
Abstract: The present invention relates to a divider for dividing a dividend by a divisor. The divider includes a subtractor for subtracting the divisor from the dividend to produce a result, storage space with a preliminary answer, and a processor for revising the dividend and preliminary answer based on the result. Each interation the divider is adapted to reiterate the subtraction and revision multiple times, based on a revised dividend and revised preliminary answer.
2 citations
TL;DR: In this article, a generalisation of the Deligne-Mumford stack with projective coarse moduli space of dimension 2 and Cartier divisors was considered, and closed-form solutions of the maximal contact log Gromov-Witten theory of the pair (Y, D) were provided for three infinite families of orbifold log Calabi-Yau surfaces.
Abstract: In [15], we established a series of correspondences relating five enumerative theories of log Calabi–Yau surfaces, i.e. pairs (Y, D) with Y a smooth projective complex surface and $$D=D_1+\dots +D_l$$
an anticanonical divisor on Y with each $$D_i$$
smooth and nef. In this paper, we explore the generalisation to Y being a smooth Deligne–Mumford stack with projective coarse moduli space of dimension 2 and $$D_i$$
nef $${\mathbb {Q}}$$
-Cartier divisors. We consider in particular three infinite families of orbifold log Calabi–Yau surfaces, and for each of them, we provide closed-form solutions of the maximal contact log Gromov–Witten theory of the pair (Y, D), the local Gromov–Witten theory of the total space of $$\bigoplus _i {\mathcal {O}}_Y(-D_i)$$
, and the open Gromov–Witten of toric orbi-branes in a Calabi–Yau 3-orbifold associated with (Y, D). We also consider new examples of BPS integral structures underlying these invariants and relate them to the Donaldson–Thomas theory of a symmetric quiver specified by (Y, D) and to a class of open/closed BPS invariants.
2 citations
Posted Content•
TL;DR: In this article, an extension of toric codes analogous to the one of Reed-Muller codes into projective ones is presented, by evaluating on the whole variety instead of considering only points with nonzero coordinates.
Abstract: Any integral convex polytope $P$ in $\mathbb{R}^N$ provides a $N$-dimensional toric variety $X_P$ and an ample divisor $D_P$ on this variety. This paper gives an explicit construction of the algebraic geometric error-correcting code on $X_P$ , obtained by evaluating global section of $\mathcal{L}(D_P)$ on every rational point of $X_P$. This work presents an extension of toric codes analogous to the one of Reed-Muller codes into projective ones, by evaluating on the whole variety instead of considering only points with non-zero coordinates. The dimension of the code is given in terms of the number of integral points in the polytope $P$ and an algorithmic technique to get a lowerbound on the minimum distance is described.
2 citations