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We give upper and lower bounds on the determinant of a perturbation of the identity matrix or, more generally, a perturbation of a nonsingular diagonal matrix. The matrices considered are, in general, diagonally dominant. The lower bounds are best possible, and in several cases they are stronger than well-known bounds due to Ostrowski and other authors. If $A = I-E$ is an $n \times n$ matrix and the elements of $E$ are bounded in absolute value by $\varepsilon \le 1/n$, then a lower bound of ...
Let $D(n)$ be the maximal determinant for $n \times n$ $\{\pm 1\}$-matrices, and ${\mathcal R}(n) = D(n)/n^{n/2}$ be the ratio of $D(n)$ to the Hadamard upper bound. We give several new lower bounds on ${\mathcal R}(n)$ in terms of $d$, where $n = h+d$, $h$ is the order of a Hadamard matrix, and $h$ is maximal subject to $h \le n$. A relatively simple bound is \[{\mathcal R}(n) \ge \left(\frac{2}{\pi e}\right)^{d/2} \left(1 - d^2\left(\frac{\pi}{2h}\right)^{1/2}\right) \;\text{ for all }\; n ...
We consider several families of binomial sum identities whose definition involves the absolute value function. In particular, we consider centered double sums of the form \[S_{\alpha,\beta}(n) := \sum_{k,\;\ell}\binom{2n}{n+k}\binom{2n}{n+\ell} |k^\alpha-\ell^\alpha|^\beta,\] obtaining new results in the cases $\alpha = 1, 2$. We show that there is a close connection between these double sums in the case $\alpha=1$ and the single centered binomial sums considered by Tuenter.
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 modified Bessel functions.
Tuenter [Fibonacci Quarterly 40 (2002), 175-180] and other authors have considered centred binomial sums of the form \[S_r(n) = \sum_k \binom{2n}{k}|n-k|^r,\] where $r$ and $n$ are non-negative integers. We consider sums of the form \[U_r(n) = \sum_k \binom{n}{k}|n/2-k|^r\] which are a generalisation of Tuenter's sums as $S_r(n) = U_r(2n)$ but $U_r(n)$ is also well-defined for odd arguments $n$. $U_r(n)$ may be interpreted as a moment of a symmetric Bernoulli random walk with $n$ steps. The f...
We present a new method for algebraic independence results in the context of Mahler's method. In particular, our method uses the asymptotic behaviour of a Mahler function $f(z)$ as $z$ goes radially to a root of unity to deduce algebraic independence results about the values of $f(z)$ at algebraic numbers. We apply our method to the canonical example of a degree two Mahler function; that is, we apply it to $F(z)$, the power series solution to the functional equation $F(z)-(1+z+z^2)F(z^4)+z^4F...
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