As Long As I Live

Photograph of Ingrid Bergman and Gary Cooper on orange backgroundhttps://scholarsjunction.msstate.edu/cht-sheet-music/8623/thumbnail.jp

Honey-Babe

Two photos of the case of Battle Cry.https://scholarsjunction.msstate.edu/cht-sheet-music/3011/thumbnail.jp

String Indexing for Top-kk Close Consecutive Occurrences

The classic string indexing problem is to preprocess a string $S$ into a compact data structure that supports efficient subsequent pattern matching queries, that is, given a pattern string $P$, report all occurrences of $P$ within $S$. In this paper, we study a basic and natural extension of string indexing called the string indexing for top-$k$ close consecutive occurrences problem (SITCCO). Here, a consecutive occurrence is a pair $(i,j)$, $i < j$, such that $P$ occurs at positions $i$ and $j$ in $S$ and there is no occurrence of $P$ between $i$ and $j$, and their distance is defined as $j-i$. Given a pattern $P$ and a parameter $k$, the goal is to report the top-$k$ consecutive occurrences of $P$ in $S$ of minimal distance. The challenge is to compactly represent $S$ while supporting queries in time close to length of $P$ and $k$. We give two time-space trade-offs for the problem. Let $n$ be the length of $S$, $m$ the length of $P$, and $\epsilon\in(0,1]$. Our first result achieves $O(n\log n)$ space and optimal query time of $O(m+k)$, and our second result achieves linear space and query time $O(m+k^{1+\epsilon})$. Along the way, we develop several techniques of independent interest, including a new translation of the problem into a line segment intersection problem and a new recursive clustering technique for trees.Comment: Fixed typos, minor change

Optimal topological simplification of discrete functions on surfaces

We solve the problem of minimizing the number of critical points among all functions on a surface within a prescribed distance {\delta} from a given input function. The result is achieved by establishing a connection between discrete Morse theory and persistent homology. Our method completely removes homological noise with persistence less than 2{\delta}, constructively proving the tightness of a lower bound on the number of critical points given by the stability theorem of persistent homology in dimension two for any input function. We also show that an optimal solution can be computed in linear time after persistence pairs have been computed.Comment: 27 pages, 8 figure

Addressable nanoantennas with cleared hotspots for single-molecule detection on a portable smartphone microscope

The advent of highly sensitive photodetectors and the development of photostabilization strategies made detecting the fluorescence of single molecules a routine task in many labs around the world. However, to this day, this process requires cost-intensive optical instruments due to the truly nanoscopic signal of a single emitter. Simplifying single-molecule detection would enable many exciting applications, e.g., in point-of-care diagnostic settings, where costly equipment would be prohibitive. Here, we introduce addressable NanoAntennas with Cleared HOtSpots (NACHOS) that are scaffolded by DNA origami nanostructures and can be specifically tailored for the incorporation of bioassays. Single emitters placed in NACHOS emit up to 461-fold (average of 89 ± 7-fold) brighter enabling their detection with a customary smartphone camera and an 8-US-dollar objective lens. To prove the applicability of our system, we built a portable, battery-powered smartphone microscope and successfully carried out an exemplary single-molecule detection assay for DNA specific to antibiotic-resistant Klebsiella pneumonia on the road

Addressable Nanoantennas with Cleared Hotspots for Single-Molecule Detection on a Portable Smartphone Microscope

The advent of highly sensitive photodetectors1,2 and the development of photostabilization strategies3 made detecting the fluorescence of a single molecule a routine task in many labs around the world. However, to this day, this process requires cost-intensive optical instruments due to the truly nanoscopic signal of a single emitter. Simplifying single-molecule detection would enable many exciting applications, e.g. in point-of-care diagnostic settings, where costly equipment would be prohibitive.4 Here, we introduce addressable NanoAntennas with Cleared HOtSpots (NACHOS) that are scaffolded by DNA origami nanostructures and can be specifically tailored for the incorporation of bioassays. Single emitters placed in the NACHOS emit up to 461-fold brighter enabling their detection with a customary smartphone camera and an 8-US-dollar objective lens. To prove the applicability of our system, we built a portable, battery-powered smartphone microscope and successfully carried out an exemplary single-molecule detection assay for DNA specific to antibiotic-resistant Klebsiella pneumonia "on the road “

Uniform convergence of discrete curvatures from nets of curvature lines

We study discrete curvatures computed from nets of curvature lines on a given smooth surface, and prove their uniform convergence to smooth principal curvatures. We provide explicit error bounds, with constants depending only on properties of the smooth limit surface and the shape regularity of the discrete net.Comment: 21 pages, 8 figure