Size reconstructibility of graphs

The deck of a graph $G$ is given by the multiset of (unlabelled) subgraphs $\{G-v:v\in V(G)\}$. The subgraphs $G-v$ are referred to as the cards of $G$. Brown and Fenner recently showed that, for $n\geq29$, the number of edges of a graph $G$ can be computed from any deck missing 2 cards. We show that, for sufficiently large $n$, the number of edges can be computed from any deck missing at most $\frac1{20}\sqrt{n}$ cards.Comment: 15 page

Nonisomorphic Ordered Sets with Arbitrarily Many Ranks That Produce Equal Decks

We prove that for any $n$ there is a pair $(P_1 ^n , P_2 ^n )$ of nonisomorphic ordered sets such that $P_1 ^n$ and $P_2 ^n$ have equal maximal and minimal decks, equal neighborhood decks, and there are $n+1$ ranks $k_0 , \ldots , k_n$ such that for each $i$ the decks obtained by removing the points of rank $k_i$ are equal. The ranks $k_1 , \ldots , k_n$ do not contain extremal elements and at each of the other ranks there are elements whose removal will produce isomorphic cards. Moreover, we show that such sets can be constructed such that only for ranks $1$ and $2$, both without extremal elements, the decks obtained by removing the points of rank $r_i$ are not equal.Comment: 30 pages, 6 figures, straight LaTe

Recovery, Renewal, and Resiliency: Gulf Coast Small Businesses Two Years Later

Presents findings from a survey of small business owners about the state of the local economy immediately following and in the two years since Katrina made landfall

A proposed DAQ system for a calorimeter at the International Linear Collider

This note describes R&D to be carried out on the data acquisition system for a calorimeter at the future International Linear Collider. A generic calorimeter and data acquisition system is described. Within this framework modified designs and potential bottlenecks within the current system are described. Solutions leading up to a technical design report will to be carried out within CALICE-UK groups.Comment: 13 pages, 4 figure

Some Ulam's reconstruction problems for quantum states

Provided a complete set of putative $k$-body reductions of a multipartite quantum state, can one determine if a joint state exists? We derive necessary conditions for this to be true. In contrast to what is known as the quantum marginal problem, we consider a setting where the labeling of the subsystems is unknown. The problem can be seen in analogy to Ulam's reconstruction conjecture in graph theory. The conjecture - still unsolved - claims that every graph on at least three vertices can uniquely be reconstructed from the set of its vertex-deleted subgraphs. When considering quantum states, we demonstrate that the non-existence of joint states can, in some cases, already be inferred from a set of marginals having the size of just more than half of the parties. We apply these methods to graph states, where many constraints can be evaluated by knowing the number of stabilizer elements of certain weights that appear in the reductions. This perspective links with constraints that were derived in the context of quantum error-correcting codes and polynomial invariants. Some of these constraints can be interpreted as monogamy-like relations that limit the correlations arising from quantum states. Lastly, we provide an answer to Ulam's reconstruction problem for generic quantum states.Comment: 22 pages, 3 figures, v2: significantly revised final versio

Visualization and Correction of Automated Segmentation, Tracking and Lineaging from 5-D Stem Cell Image Sequences

Results: We present an application that enables the quantitative analysis of multichannel 5-D (x, y, z, t, channel) and large montage confocal fluorescence microscopy images. The image sequences show stem cells together with blood vessels, enabling quantification of the dynamic behaviors of stem cells in relation to their vascular niche, with applications in developmental and cancer biology. Our application automatically segments, tracks, and lineages the image sequence data and then allows the user to view and edit the results of automated algorithms in a stereoscopic 3-D window while simultaneously viewing the stem cell lineage tree in a 2-D window. Using the GPU to store and render the image sequence data enables a hybrid computational approach. An inference-based approach utilizing user-provided edits to automatically correct related mistakes executes interactively on the system CPU while the GPU handles 3-D visualization tasks. Conclusions: By exploiting commodity computer gaming hardware, we have developed an application that can be run in the laboratory to facilitate rapid iteration through biological experiments. There is a pressing need for visualization and analysis tools for 5-D live cell image data. We combine accurate unsupervised processes with an intuitive visualization of the results. Our validation interface allows for each data set to be corrected to 100% accuracy, ensuring that downstream data analysis is accurate and verifiable. Our tool is the first to combine all of these aspects, leveraging the synergies obtained by utilizing validation information from stereo visualization to improve the low level image processing tasks.Comment: BioVis 2014 conferenc

Frequency-modulated continuous-wave LiDAR compressive depth-mapping

We present an inexpensive architecture for converting a frequency-modulated continuous-wave LiDAR system into a compressive-sensing based depth-mapping camera. Instead of raster scanning to obtain depth-maps, compressive sensing is used to significantly reduce the number of measurements. Ideally, our approach requires two difference detectors. % but can operate with only one at the cost of doubling the number of measurments. Due to the large flux entering the detectors, the signal amplification from heterodyne detection, and the effects of background subtraction from compressive sensing, the system can obtain higher signal-to-noise ratios over detector-array based schemes while scanning a scene faster than is possible through raster-scanning. %Moreover, we show how a single total-variation minimization and two fast least-squares minimizations, instead of a single complex nonlinear minimization, can efficiently recover high-resolution depth-maps with minimal computational overhead. Moreover, by efficiently storing only $2m$ data points from $m<n$ measurements of an $n$ pixel scene, we can easily extract depths by solving only two linear equations with efficient convex-optimization methods