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The Brent-McMillan algorithm B3 (1980), when implemented with binary splitting, is the fastest known algorithm for high-precision computation of Euler's constant. However, no rigorous error bound for the algorithm has ever been published. We provide such a bound and justify the empirical observations of Brent and McMillan. We also give bounds on the error in the asymptotic expansions of functions related to the Bessel functions I0(x) and K0(x) for positive real x. [ABSTRACT FROM AUTHOR]

In this paper, we prove that for any positive fundamental discriminant D > 1596, there is always at least one prime p ≤ D0.45 such that the Kronecker symbol (D/p) = -1. This improves a result of Granville, Mollin and Williams, where they showed that the least inert prime p in a real quadratic field of discriminant D > 3705 is at most √D/2. We use a "smoothed" version of the P´olya-Vinogradov inequality, which is very useful for numerically explicit estimates. [ABSTRACT FROM AUTHOR]

For relatively prime positive integers $u_0$ and $r$, we consider the arithmetic progression ${u_k:=u_0+kr}_{k=0}^n$. par Define $L_n:=operatorname {lcm}{u_0, u_1, ..., u_n}$ and let $age 2$ be any integer. In this paper, we show that for integers $alpha , rgeq a$ and $ngeq 2alpha r$, we have $$L_ngeq u_0r^{alpha +a-2}(r+1)^n.$$ In particular, letting $a=2$ yields an improvement to the best previous lower bound on $L_n$ (obtained by Hong and Yang) for all but three choices of $alpha , rgeq 2$. [ABSTRACT FROM AUTHOR]

The $w$-index of a surface link $F$ is the minimal number of the triple points of surface braids representing $F$. In this paper, for a given $3$-cocycle, we consider the minimal number of the $w$-indices of surface links whose quandle cocycle invariants associated with $f$ are non-trivial, and denote it $omega (f)$. In particular, we show that $omega (theta _3)=6$ and $omega (theta _p)geq 7$, where $theta _n$ is Mochizuki's $3$-cocycle of the dihedral quandle of order $n$ and $p$ is an odd prime integer $not =3$. As a consequence, for a given non-negative integer $g$, there are surface knots with genus $g$ with the $w$-index $6$. [ABSTRACT FROM AUTHOR]

Given a function f : E → R with E ⊂ &#x211D:n, we explain how to decide whether f extends to a ⊂m function F on ℝn. If E is finite, then one can efficiently compute an F as above, whose Cm norm has the least possible order of magnitude (joint work with B. Klartag). [ABSTRACT FROM AUTHOR]

In this paper we discuss the behavior of the Gromov boundaries and limit sets for the surface subgroups of the mapping class group with accidental parabolics constructed by the author and A. Reid (2006). Specifically, we show that generically there are no Cannon--Thurston maps from the Gromov boundary to Thurston's boundary of Teichmüller space. [ABSTRACT FROM AUTHOR]