arXiv subjects

David Lewis

Publications and source records attributed to David Lewis.

21 records · Page 2Linked to original sources

arXiv2020

On Covering Numbers, Young Diagrams, and the Local Dimension of Posets

We study covering numbers and local covering numbers with respect to difference graphs and complete bipartite graphs. In particular we show that in every cover of a Young diagram with $\binom{2k}{k}$ steps with generalized rectangles there is a row or a column in the diagram that is used by at least $k+1$ rectangles, and prove that this is best-possible. This answers two questions by Kim, Martin, Masa{ř}{\'ı}k, Shull, Smith, Uzzell, and Wang (Europ. J. Comb. 2020), namely: - What is the local complete bipartite cover number of a difference graph? - Is there a sequence of graphs with constant local difference graph cover number and unbounded local complete bipartite cover number? We add to the study of these local covering numbers with a lower bound construction and some examples. Following Kim \emph{et al.}, we use the results on local covering numbers to provide lower and upper bounds for the local dimension of partially ordered sets of height~2. We discuss the local dimension of some posets related to Boolean lattices and show that the poset induced by the first two layers of the Boolean lattice has local dimension $(1 + o(1))\log_2\log_2 n$. We conclude with some remarks on covering numbers for digraphs and Ferrers dimension.

math.CO↗

arXiv2019

Rainbow saturation of graphs

In this paper we study the following problem proposed by Barrus, Ferrara, Vandenbussche, and Wenger. Given a graph $H$ and an integer $t$, what is $\operatorname{sat}_{t}\left(n, \mathfrak{R}{(H)}\right)$, the minimum number of edges in a $t$-edge-coloured graph $G$ on $n$ vertices such that $G$ does not contain a rainbow copy of $H$, but adding to $G$ a new edge in any colour from $\{1,2,\ldots,t\}$ creates a rainbow copy of $H$? Here, we completely characterize the growth rates of $\operatorname{sat}_{t}\left(n, \mathfrak{R}{(H)}\right)$ as a function of $n$, for any graph $H$ belonging to a large class of connected graphs and for any $t\geq e(H)$. This classification includes all connected graphs of minimum degree $2$. In particular, we prove that $\operatorname{sat}_{t}\left(n, \mathfrak{R}{(K_r)}\right)=Θ(n\log n)$, for any $r\geq 3$ and $t\geq {r \choose 2}$, thus resolving a conjecture of Barrus, Ferrara, Vandenbussche, and Wenger. We also pose several new problems and conjectures.

math.CO↗

arXiv2016

Quantum State Absorptions Coupled To Resonance Raman Spectroscopy Could Result In A General Explanation of TERS

Tip enhanced Raman scattering (TERS) amplifies the intensity of vibrational Raman scattering by employing the tip of a probe interacting, in ultra close proximity, with a surface. Although a general understanding of the TERS process is still to be fully elucidated, scanning tunneling microscopy (STM) feedback is often applied with success in TERS to keep a noble metal probe in intimate proximity with a noble metal substrate. Since such STM TERS is a common modality, the possible implications of plasmonic fields that may be induced by the tunneling process are investigated and reported. In addition, TERS of a 2D resonant molecular system, a MoS2 bilayer crystal and a 2D non-resonant, lipid molecular bilayer is compared. Data with multiple excitation wavelengths and surfaces for the resonant system in the near- (TERS) and far-field regimes are reported. An interpretation based on weak coupling interactions within the framework of conventional resonance Raman scattering can explain the observed TERS enhancements. The non-resonant molecular lipid system, on the other hand, requires strong coupling for a full understanding of the reported observations.

physics.optics↗