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Discovery Projects - Grant ID: DP140101417

Title
Discovery Projects - Grant ID: DP140101417
Funding
ARC | Discovery Projects
Contract (GA) number
DP140101417
Start Date
2014/01/01
End Date
2016/12/31
Open Access mandate
no
Organizations
-
More information
http://purl.org/au-research/grants/arc/DP140101417

 

  • Some binomial sums involving absolute values

    Brent, Richard P.; Ohtsuka, Hideyuki; Osborn, Judy-anne H.; Prodinger, Helmut (2014)
    Projects: ARC | Discovery Projects - Grant ID: DP140101417 (DP140101417)
    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.

    Computation of Maximal Determinants of Binary Circulant Matrices

    Brent, Richard P.; Yedidia, Adam B. (2018)
    Projects: ARC | Discovery Projects - Grant ID: DP140101417 (DP140101417)
    We describe algorithms for computing maximal determinants of binary circulant matrices of small orders. Here "binary matrix" means a matrix whose elements are drawn from $\{0,1\}$ or $\{-1,1\}$. We describe efficient parallel algorithms for the search, using Duval's algorithm for generation of necklaces and the well-known representation of the determinant of a circulant in terms of roots of unity. Tables of maximal determinants are given for orders $\le 52$. Our computations extend earlier re...

    Asymptotic approximation of central binomial coefficients with rigorous error bounds

    We show that a well-known asymptotic series for the logarithm of the central binomial coefficient is strictly enveloping, so the error incurred in truncating the series is of the same sign as the next term, and is bounded in magnitude by that term. We also consider some closely related asymptotic series, and make some historical remarks.

    Bounds on determinants of perturbed diagonal matrices

    Brent, Richard P.; Osborn, Judy-anne H.; Smith, Warren D. (2014)
    Projects: ARC | Discovery Projects - Grant ID: DP140101417 (DP140101417)
    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 ...

    Twelve new primitive binary trinomials

    Brent, Richard P.; Zimmermann, Paul (2016)
    Projects: ARC | Discovery Projects - Grant ID: DP140101417 (DP140101417)
    We exhibit twelve new primitive trinomials over GF(2) of record degrees 42 643 801, 43 112 609, and 74 207 281. In addition we report the first Mersenne exponent not ruled out by Swan's theorem [10] — namely 57 885 161 — for which none primitive trinomial exists. This completes the search for the currently known Mersenne prime exponents.

    On asymptotic approximations to the log-Gamma and Riemann-Siegel theta functions

    We give bounds on the error in the asymptotic approximation of the log-Gamma function $\ln\Gamma(z)$ for complex $z$ in the right half-plane. These improve on earlier bounds by Behnke and Sommer (1962), Spira (1971), and Hare (1997). We show that $|R_{k+1}(z)/T_k(z)| < \sqrt{\pi k}$ for nonzero $z$ in the right half-plane, where $T_k(z)$ is the $k$-th term in the asymptotic series, and $R_{k+1}(z)$ is the error incurred in truncating the series after $k$ terms. If $k \le |z|$, then the strong...

    The Borwein brothers, Pi and the AGM

    We consider some of Jonathan and Peter Borweins' contributions to the high-precision computation of $\pi$ and the elementary functions, with particular reference to their book "Pi and the AGM" (Wiley, 1987). Here "AGM" is the arithmetic-geometric mean of Gauss and Legendre. Because the AGM converges quadratically, it can be combined with fast multiplication algorithms to give fast algorithms for the {$n$-bit} computation of $\pi$, and more generally the elementary functions. These algorithms ...

    Probabilistic lower bounds on maximal determinants of binary matrices

    Brent, Richard P.; Osborn, Judy-anne H.; Smith, Warren D. (2015)
    Projects: ARC | Discovery Projects - Grant ID: DP140101417 (DP140101417)
    Let ${\mathcal D}(n)$ be the maximal determinant for $n \times n$ $\{\pm 1\}$-matrices, and $\mathcal R(n) = {\mathcal D}(n)/n^{n/2}$ be the ratio of ${\mathcal D}(n)$ to the Hadamard upper bound. Using the probabilistic method, we prove new lower bounds on ${\mathcal D}(n)$ and $\mathcal R(n)$ in terms of $d = n-h$, where $h$ is the order of a Hadamard matrix and $h$ is maximal subject to $h \le n$. For example, $\mathcal R(n) > (\pi e/2)^{-d/2}$ if $1 \le d \le 3$, and $\mathcal R(n) > (\pi...

    Lower bounds on maximal determinants of binary matrices via the probabilistic method

    Brent, Richard P.; Osborn, Judy-anne H.; Smith, Warren D. (2014)
    Projects: ARC | Discovery Projects - Grant ID: DP140101417 (DP140101417)
    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 ...

    Generalising Tuenter's binomial sums

    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...
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