On minors of maximal determinant matrices

By an old result of Cohn (1965), a Hadamard matrix of order n has no proper Hadamard submatrices of order m > n/2. We generalise this result to maximal determinant submatrices of Hadamard matrices, and show that an interval of length asymptotically equal to n/2 is excluded from the allowable orders. We make a conjecture regarding a lower bound for sums of squares of minors of maximal determinant matrices, and give evidence in support of the conjecture. We give tables of the values taken by the minors of all maximal determinant matrices of orders up to and including 21 and make some observations on the data. Finally, we describe the algorithms that were used to compute the tables.Comment: 35 pages, 43 tables, added reference to Cohn in v

Probabilistic lower bounds on maximal determinants of binary matrices

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 $\mathcal R(n) > (\pi e/2)^{-d/2}(1 - d^2(\pi/(2h))^{1/2})$ if $d > 3$. By a recent result of Livinskyi, $d^2/h^{1/2} \to 0$ as $n \to \infty$, so the second bound is close to $(\pi e/2)^{-d/2}$ for large $n$. Previous lower bounds tended to zero as $n \to \infty$ with $d$ fixed, except in the cases $d \in \{0,1\}$. For $d \ge 2$, our bounds are better for all sufficiently large $n$. If the Hadamard conjecture is true, then $d \le 3$, so the first bound above shows that $\mathcal R(n)$ is bounded below by a positive constant $(\pi e/2)^{-3/2} > 0.1133$.Comment: 17 pages, 2 tables, 24 references. Shorter version of arXiv:1402.6817v4. Typos corrected in v2 and v3, new Lemma 7 in v4, updated references in v5, added Remark 2.8 and a reference in v6, updated references in v

Some binomial sums involving absolute values

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.Comment: 15 pages, 19 reference

Bounds on minors of binary matrices

We prove an upper bound on sums of squares of minors of {+1, -1} matrices. The bound is sharp for Hadamard matrices, a result due to de Launey and Levin (2009), but our proof is simpler. We give several corollaries relevant to minors of Hadamard matrices, and generalise a result of Turan on determinants of random {+1,-1} matrices.Comment: 9 pages, 1 table. Typo corrected in v2. Two references and Theorem 2 added in v

Adaptive assessment for differing maths backgrounds?

We report upon a work in progress: an attempt to use adaptive features of online mathematics homework system MapleTA to address educational challenges associated with diverse backgrounds of students in a large first year mathematics course. The course is both a service course taken by several hundred Engineering and Science students, as well as the first course for a significant proportion of Bachelor of Mathematics students. Our rationale for the MapleTA assignments is that students of different backgrounds learn mathematics at different speeds: this is difficult to cater for in a didactic lecture, but may be better addressed in online homework which is structured to give varying levels of scaffolding depending upon the live responses of each individual student. Our medium-term hope is to use such assignments as the basis for a "Flipped Classroom" approach, in which students first encounter course material outside of class time, and then class-time is used to discuss and digest those aspects students are finding difficult. Our approach is supported by the literature, which shows that active learning contexts and Computer Assisted Instruction (CAI) can both be helpful in promoting learning and improving retention, but that they need to be implemented carefully. Our current implementation is of partially adaptive assignments that are run in addition to largely didactic lectures. We will also briefly discuss our use of multiple-choice "card questions" as adapted from the "Peer Instruction" work of Harvard Physicist Professor Eric Mazur, to promote more discursive mathematics lectures. The focus of the presentation will be upon the MapleTA assignments themselves and the degree of adaptivity that we have been able to implement thus far, together with a description of some of the practical pitfalls along the way, for those interested in trialling a similar approach

Binary Constant-Length Substitutions and Mahler Measures of Borwein Polynomials

Baake M, Coons M, Mañibo CN. Binary Constant-Length Substitutions and Mahler Measures of Borwein Polynomials. In: Bailey DH, Borwein NS, Brent RP, et al., eds. From Analysis to Visualization : A Celebration of the Life and Legacy of Jonathan M. Borwein, Callaghan, Australia, September 2017. Springer Proceedings in Mathematics & Statistics. Vol 313. Cham: Springer International Publishing; 2020: 303-322

Coronal Heating as Determined by the Solar Flare Frequency Distribution Obtained by Aggregating Case Studies

Flare frequency distributions represent a key approach to addressing one of the largest problems in solar and stellar physics: determining the mechanism that counter-intuitively heats coronae to temperatures that are orders of magnitude hotter than the corresponding photospheres. It is widely accepted that the magnetic field is responsible for the heating, but there are two competing mechanisms that could explain it: nanoflares or Alfv\'en waves. To date, neither can be directly observed. Nanoflares are, by definition, extremely small, but their aggregate energy release could represent a substantial heating mechanism, presuming they are sufficiently abundant. One way to test this presumption is via the flare frequency distribution, which describes how often flares of various energies occur. If the slope of the power law fitting the flare frequency distribution is above a critical threshold, $\alpha=2$ as established in prior literature, then there should be a sufficient abundance of nanoflares to explain coronal heating. We performed $>$600 case studies of solar flares, made possible by an unprecedented number of data analysts via three semesters of an undergraduate physics laboratory course. This allowed us to include two crucial, but nontrivial, analysis methods: pre-flare baseline subtraction and computation of the flare energy, which requires determining flare start and stop times. We aggregated the results of these analyses into a statistical study to determine that $\alpha = 1.63 \pm 0.03$. This is below the critical threshold, suggesting that Alfv\'en waves are an important driver of coronal heating.Comment: 1,002 authors, 14 pages, 4 figures, 3 tables, published by The Astrophysical Journal on 2023-05-09, volume 948, page 7