Songwriting with Iconic Notation in a Music Technology Classroom

Recognizing that music teachers may struggle to implement songwriting activities in a classroom, and that iconic notation provides an opportunity to increase access to school music for all students, the purpose of this article is to share one model of songwriting activities in a music technology class using chord diagrams, beat grids, and keyboard charts. The article outlines specific steps to the creation of drum grooves, simple chord progressions, bass lines, and melodies, using forms of notation that are appropriate for popular music instruments and styles.

Unfolding Cubes: Nets, Packings, Partitions, Chords

We show that every ridge unfolding of an $n$-cube is without self-overlap, yielding a valid net. The results are obtained by developing machinery that translates cube unfolding into combinatorial frameworks. Moreover, the geometry of the bounding boxes of these cube nets are classified using integer partitions, as well as the combinatorics of path unfoldings seen through the lens of chord diagrams.

The double scaled limit of super-symmetric SYK models

We compute the exact density of states and 2-point function of the $$ \mathcal{N} $$ N = 2 super-symmetric SYK model in the large N double-scaled limit, by using combinatorial tools that relate the moments of the distribution to sums over oriented chord diagrams. In particular we show how SUSY is realized on the (highly degenerate) Hilbert space of chords. We further calculate analytically the number of ground states of the model in each charge sector at finite N, and compare it to the results from the double-scaled limit. Our results reduce to the super-Schwarzian action in the low energy short interaction length limit. They imply that the conformal ansatz of the 2-point function is inconsistent due to the large number of ground states, and we show how to add this contribution. We also discuss the relation of the model to SLq(2|1). For completeness we present an overview of the $$ \mathcal{N} $$ N = 1 super-symmetric SYK model in the large N double-scaled limit.

Crosscap number of knots and volume bounds

In this paper, we obtain the crosscap number of any alternating knots by using our recently-introduced diagrammatic knot invariant (Theorem 1). The proof is given by properties of chord diagrams (Kindred proved Theorem 1 independently via other techniques). For non-alternating knots, we give Theorem 2 that generalizes Theorem 1. We also improve known formulas to obtain upper bounds of the crosscap number of knots (alternating or non-alternating) (Theorem 3). As a corollary, this paper connects crosscap numbers and our invariant with other knot invariants such as the Jones polynomial, twist number, crossing number, and hyperbolic volume (Corollaries 1–7). In Appendix A, using Theorem 1, we complete giving the crosscap numbers of the alternating knots with up to 11 crossings including those of the previously unknown values for [Formula: see text] knots (Tables A.1).

Chaos on the hypercube

We analyze the spectral properties of a d-dimensional HyperCubic (HC) lattice model originally introduced by Parisi. The U(1) gauge links of this model give rise to a magnetic flux of constant magnitude ϕ but random orientation through the faces of the hypercube. The HC model, which also can be written as a model of 2d interacting Majorana fermions, has a spectral flow that is reminiscent of Maldacena-Qi (MQ) model, and its spectrum at ϕ = 0, actually coincides with the coupling term of the MQ model. As was already shown by Parisi, at leading order in 1/d, the spectral density of this model is given by the density function of the Q-Hermite polynomials, which is also the spectral density of the double-scaled Sachdev-Ye-Kitaev model. Parisi demonstrated this by mapping the moments of the HC model to Q-weighted sums on chord diagrams. We point out that the subleading moments of the HC model can also be mapped to weighted sums on chord diagrams, in a manner that descends from the leading moments. The HC model has a magnetic inversion symmetry that depends on both the magnitude and the orientation of the magnetic flux through the faces of the hypercube. The spectrum for fixed quantum number of this symmetry exhibits a transition from regular spectra at ϕ = 0 to chaotic spectra with spectral statistics given by the Gaussian Unitary Ensembles (GUE) for larger values of ϕ. For small magnetic flux, the ground state is gapped and is close to a Thermofield Double (TFD) state.

MultiGWAS: An integrative tool for Genome Wide Association Studies (GWAS) in tetraploid organisms

SummaryThe Genome-Wide Association Studies (GWAS) are essential to determine the genetic bases of either ecological or economic phenotypic variation across individuals within populations of model and non-model organisms. For this research question, current practice is the replication of the GWAS testing different parameters and models to validate the reproducibility of results. However, straightforward methodologies that manage both replication and tetraploid data are still missing. To solve this problem, we designed the MultiGWAS, a tool that does GWAS for diploid and tetraploid organisms by executing in parallel four software, two for polyploid data (GWASpoly and SHEsis) and two for diploids data (PLINK and TASSEL). MultiGWAS has several advantages. It runs either in the command line or in an interface. It manages different genotype formats, including VCF. It executes both the full and naïve models using several quality filters. Besides, it calculates a score to choose the best gene action model across GWASPoly and TASSEL. Finally, it generates several reports that facilitate the identification of false associations from both the significant and the best-ranked association SNP among the four software. We tested MultiGWAS with tetraploid potato data. The execution demonstrated that the Venn diagram and the other companion reports (i.e., Manhattan and QQ plots, heatmaps for associated SNP profiles, and chord diagrams to trace associated SNP by chromosomes) were useful to identify associated SNP shared among different models and parameters. Therefore, we confirmed that MultiGWAS is a suitable wrapping tool that successfully handles GWAS replication in both diploid and tetraploid [email protected]