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.

Mordell-Weil Torsion and the Global Structure of Gauge Groups in F-theory

Abstract: We study the global structure of the gauge group $G$ of F-theory compactified on an elliptic fibration $Y$. The global properties of $G$ are encoded in the torsion subgroup of the Mordell-Weil group of rational sections of $Y$. Generalising the Shioda map to torsional sections we construct a specific integer divisor class on $Y$ as a fractional linear combination of the resolution divisors associated with the Cartan subalgebra of $G$. This divisor class can be interpreted as an element of the refined coweight lattice of the gauge group. As a result, the spectrum of admissible matter representations is strongly constrained and the gauge group is non-simply connected. We exemplify our results by a detailed analysis of the general elliptic fibration with Mordell-Weil group $\mathbb Z_2$ and $\mathbb Z_3$ as well as a further specialization to $\mathbb Z \oplus \mathbb Z_2$. Our analysis exploits the representation of these fibrations as hypersurfaces in toric geometry.

On the normalization requirement of divisor in divide-and-correct methods

Abstract: This paper presents an analysis on the normalization requirement of the divisor in a divide-and-correct method. This analysis is made subject to the condition that not more than one correction is required to obtain the true quotient character, from the trial estimate got from the division of a two-precision segment of every partial remainder by a suitably rounded single-precision divisor. (This segmented division is denoted here as a (2, 1) precision basic division.) It is found that the normalization requirement could be narrowed down to a smaller range of divisors, provided the magnitude of the character next to the leading character of the divisor is known. If, however, the normalization is to be eliminated one has to choose proper higher precision segments of operands for the basic division.Also considered is the possibility of eliminating the normalization by an increase on the number of corrections on the quotient estimate got from a (2, 1) precision basic division. It is shown that such a scheme is economical only for small radices.

Division by constant for the ST100 DSP microprocessor

Abstract: Algorithms for Euclidean (i.e., integer) division by a constant operation are presented. They allow fast computation for some values of the divisor (known at compile time) or also when both quotient and modulus are required. These algorithms are based on the multiply-accumulate instruction and the 40-bit arithmetic available in DSPs such as the ST100 DSP from STMicroelectronics. The results are demonstrated in the case of standard speech coding applications.

Lower bounds for the variance of sequences in arithmetic progressions: primes and divisor functions

Abstract: We develop a general method for lower bounding the variance of sequences in arithmetic progressions mod $q$, summed over all $q \leq Q$, building on previous work of Liu, Perelli, Hooley, and others. The proofs lower bound the variance by the minor arc contribution in the circle method, which we lower bound by comparing with suitable auxiliary exponential sums that are easier to understand. As an application, we prove a lower bound of $(1-\epsilon) QN\log(Q^2/N)$ for the variance of the von Mangoldt function $(\Lambda(n))_{n=1}^{N}$, on the range $\sqrt{N} (\log N)^C \leq Q \leq N$. Previously such a result was only available assuming the Riemann Hypothesis. We also prove a lower bound $\gg_{k,\delta} Q N (\log N)^{k^2 - 1}$ for the variance of the divisor functions $d_k(n)$, valid on the range $N^{1/2+\delta} \leq Q \leq N$, for any natural number $k \geq 2$.