arXiv subjects
Publications and source records attributed to Derek Wu.
A tensor of degenerate geometric rank corresponds to a linear space of matrices in which some locus of matrices of rank at most $r$ has unexpectedly large dimension. We prove a sufficient criterion for when such a tensor cannot be lifted to a larger tensor and characterize those tensors for which the degeneracy is caused by the rank $r=1$ locus. We also provide new examples of tensors with degenerate geometric rank stemming from nonlinear loci.
A fundamental identity for the number of vacillating tableaux was originally obtained through the representation theory of partition algebras. We extend this identity to arbitrary differential posets and show that it, together with several analogous identities, follows directly from the structural properties of differential posets. Specializing to Young's lattice and its Cartesian powers, we further obtain new bijective proofs via a simple deletion--insertion process.
We prove border rank bounds for a class of $GL(V)$-invariant tensors in $V^*\otimes U\otimes W$, where $U$ and $W$ are $GL(V)$-modules. These tensors correspond to spaces of matrices of constant rank. In particular we prove lower bounds for tensors in $\mathbb{C}^l\otimes\mathbb{C}^m\otimes\mathbb{C}^n$ that are not $1_A$-generic, where no nontrivial bounds were known, and also when $l,m\ll n$, where previously only bounds for unbalanced matrix multiplication tensors were known. We give the first explicit use of Young flattenings for tensors beyond Koszul to obtain border rank lower bounds, and determine the border rank of three tensors.
With higher frequencies and broader spectrum than conventional frequency bands, the millimeter-wave (mmWave) band is suitable for next-generation wireless networks featuring short-distance high-rate communications. As a newcomer, mmWaves are expected to have the backward compatibility with existing services and collaborate with other technologies in order to enhance system performances. Therefore, the coexistence issues become an essential topic for next-generation wireless communications. In this paper, we systematically review the coexistence issues of broadband mmWave communications and their corresponding solutions proposed in the literature, helping shed light on the insights of the mmWave design. Particularly, the works surveyed in this paper can be classified into four categories: coexistence with microwave communications, coexistence with fixed services, coexistence with non-orthogonal multiple access (NOMA), and other coexistence issues. Results of numerical evaluations inspired by the literature are presented for a deeper analysis. We also point out some challenges and future directions for each category as a roadmap to further investigate the coexistence issues of broadband mmWave communications.
We use Young tableaux to compute the dimension of $V^r$, the Prym-Brill-Noether locus of a folded chain of loops of any gonality. This tropical result yields a new upper bound on the dimensions of algebraic Prym-Brill-Noether loci. Moreover, we prove that $V^r$ is pure-dimensional and connected in codimension $1$ when $\dim V^r \geq 1$. We then compute the first Betti number of this locus for even gonality when the dimension is exactly $1$, and compute the cardinality when the locus is finite and the edge lengths are generic.