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A big challenge in current systems biology research arises when different types of data must be accessed from separate sources and visualized using separate tools. The high cognitive load required to navigate such a workflow is detrimental to hypothesis generation. Accordingly, there is a need for a robust research platform that incorporates all data and provides integrated search, analysis, and visualization features through a single portal. Here, we present ePlant (http://bar.utoronto.ca/eplant), a visual analytic tool for exploring multiple levels of Arabidopsis thaliana data through a zoomable user interface. ePlant connects to several publicly available web services to download genome, proteome, interactome, transcriptome, and 3D molecular structure data for one or more genes or gene products of interest. Data are displayed with a set of visualization tools that are presented using a conceptual hierarchy from big to small, and many of the tools combine information from more than one data type. We describe the development of ePlant in this article and present several examples illustrating its integrative features for hypothesis generation. We also describe the process of deploying ePlant as an “app” on Araport. Building on readily available web services, the code for ePlant is freely available for any other biological species research.

Major urinary proteins (MUPs) are often suggested to be highly polymorphic, and thereby provide unique chemical signatures used for individual and genetic kin recognition; however, studies on MUP variability have been lacking. We surveyed populations of wild house mice ( Mus musculus musculus ), and examined variation of MUP genes and proteins. We sequenced several Mup genes (9 to 11 loci) and unexpectedly found no inter-individual variation. We also found that microsatellite markers inside the MUP cluster show remarkably low levels of allelic diversity, and significantly lower than the diversity of markers flanking the cluster or other markers in the genome. We found low individual variation in the number and types of MUP proteins using a shotgun proteomic approach, even among mice with variable MUP electrophoretic profiles. We identified gel bands and spots using high-resolution mass spectrometry and discovered that gel-based methods do not separate MUP proteins, and therefore do not provide measures of MUP diversity, as generally assumed. The low diversity and high homology of Mup genes are likely maintained by purifying selection and gene conversion, and our results indicate that the type of selection on MUPs and their adaptive functions need to be re-evaluated.

In \\(d\\)-dimensional space (over any field), given a set of lines, a joint is a point passed through by \\(d\\) lines not all lying in some hyperplane. The joints problem asks to determine the maximum number of joints formed by \\(L\\) lines, and it was one of the successes of the Guth--Katz polynomial method. We prove a new upper bound on the number of joints that matches, up to a \\(1+o(1)\\) factor, the best known construction: place \\(k\\) generic hyperplanes, and use their \\((d-1)\\)-wise intersections to form \\(\\binom{k}{d-1}\\) lines and their \\(d\\)-wise intersections to form \\(\\binom{k}{d}\\) joints. Guth conjectured that this construction is optimal. Our technique builds on the work on Ruixiang Zhang proving the multijoints conjecture via an extension of the polynomial method. We set up a variational problem to control the high order of vanishing of a polynomial at each joint.

In this paper, we investigate several extremal combinatorics problems that ask for the maximum number of copies of a fixed subgraph given the number of edges. We call problems of this type Kruskal--Katona-type problems. Most of the problems that will be discussed in this paper are related to the joints problem. There are two main results in this paper. First, we prove that, in a \\(3\\)-edge-colored graph with \\(R\\) red, \\(G\\) green, \\(B\\) blue edges, the number of rainbow triangles is at most \\(\\sqrt{2RGB}\\), which is sharp. Second, we give a generalization of the Kruskal--Katona theorem that implies many other previous generalizations. Both arguments use the entropy method, and the main innovation lies in a more clever argument that improves bounds given by Shearer's inequality.

A self-organizing network refers to a computer network that can configure, manage, and optimize itself. To achieve this goal, it first collects the network information for performance analysis at some control entities. After the control entities determine the optimized parameters, they push the settings back to the network again using the same channel they use to collect the information. Network optimization has been extensively studied over the last two decades. Most works ignore actual mechanisms for data collection and parameter update. This is partially true because device vendors, such as Cisco or Juniper, have their own control interfaces. Unfortunately, these interfaces are generally not compatible with each other, and so a unified control protocol, OpenFlow, emerged. OpenFlow defines a set of commands for data collection and parameter update for wired networks. However, wireless networks, or the most richly existing Wi-Fi, were not the primary targets of OpenFlow. In this dissertation, we first showed OpenFlow protocol could be migrated from wired networks to wireless networks with minor software modification and use to build a software-defined mobile ad hoc network (SD MANET) prototype. Seeing the tight constraints in our SD MANET, we moved on to loosen these limitations. We came up with an approach to support most nowadays smart devices using the standard IEEE 802.11v/r protocol to replace the last mile connection between the devices and their associated access points. Our solution did not require any hardware or software modification on users' devices but only need the built-in IEEE 802.11v/r support, which can be found in many mainstream smart devices. In addition to smart devices having rich network capability. We noticed another group of wireless connectivity devices but did not support advanced protocols such as IEEE 802.11v/r due to their simplified architectures. These devices include the ever-popular internet-of-things devices built on simple, low-power and low-cost Wi-Fi system-on-chip solutions. Because Wi-Fi connection consumes lots of power, today's Wi-Fi IoT devices generally require external power support, and we can hardly see battery-powered Wi-Fi IoT devices. Seeing the demand, we design a non-coherent wake-up receiver that takes over the channel monitoring task of a power-consuming Wi-Fi interface so that the Wi-Fi interface can be completely turned off. Our wake-up receiver consumes only 20-40 $\\mu$W when monitoring the channel, whereas a general Wi-Fi interface can easily drain more than 100 mW. We also came up with an extendable finite-state-machine design that supports multicast wake-up. Multiple receivers can wake up through a single, carefully selected wake-up signal. Practicality is the main idea of this dissertation. We have seen solutions with amazing performance but required either huge investment or complicated hardware design. In this dissertation, we chose the other way around by first analyzing the capabilities of existing frameworks and design solutions installed as overlays. Through this process, practicality is guaranteed.

We generalize the Guth--Katz joints theorem from lines to varieties. A special case says that \\(N\\) planes (2-flats) in 6 dimensions (over any field) have \\(O(N^{3/2})\\) joints, where a joint is a point contained in a triple of these planes not all lying in some hyperplane. More generally, we prove the same bound when the set of \\(N\\) planes is replaced by a set of 2-dimensional algebraic varieties of total degree \\(N\\), and a joint is a point that is regular for three varieties whose tangent planes at that point are not all contained in some hyperplane. Our most general result gives upper bounds, tight up to constant factors, for joints with multiplicities for several sets of varieties of arbitrary dimensions (known as Carbery's conjecture). Our main innovation is a new way to extend the polynomial method to higher dimensional objects, relating the degree of a polynomial and its orders of vanishing on a given set of points on a variety.

A sock ordering is a sequence of socks with different colors. A sock ordering is foot-sortable if the sequence of socks can be sorted by a stack so that socks with the same color form a contiguous block. The problem of deciding whether a given sock ordering is foot-sortable was first considered by Defant and Kravitz, who resolved the case for alignment-free 2-uniform sock orderings. In this paper, we resolve the problem in a more general setting, where each color appears in the sock ordering at most twice. A key component of the argument is a fast algorithm that determines the foot-sortability of a sock ordering of length \\(N\\) in time \\(O(N\\log N)\\), which is also an interesting result on its own.

In this paper, we provide a new proof of a density version of Turán's theorem. We also rephrase both the theorem and the proof using entropy. With the entropic formulation, we show that some naturally defined entropic quantity is closely connected to other common quantities such as Lagrangian and spectral radius. In addition, we also determine the Turán density for a new family of hypergraphs, which we call tents. Our result can be seen as a new generalization of Mubayi's result on the extended cliques.

We prove three main results about semialgebraic hypergraphs. First, we prove an optimal and oblivious regularity lemma. Fox, Pach, and Suk proved that the class of \\(k\\)-uniform semialgebraic hypergraphs satisfies a very strong regularity lemma where the vertex set can be partitioned into \\(\\mathrm{poly}(1/\\varepsilon)\\) parts so that all but an \\(\\varepsilon\\)-fraction of \\(k\\)-tuples of parts are homogeneous (either complete or empty). Our result improves the number of parts in the partition to \\(O_{d,k}((D/\\varepsilon)^{d})\\) where \\(d\\) is the dimension of the ambient space and \\(D\\) is a measure of the complexity of the hypergraph; additionally, the partition is oblivious to the edge set of the hypergraph. We give examples that show that the dependence on both \\(\\varepsilon\\) and \\(D\\) is optimal. From this regularity lemma we deduce the best-known Turán-type result for semialgebraic hypergraphs. Third, we prove a Zarankiewicz-type result for semialgebraic hypergraphs. Previously Fox, Pach, Sheffer, Suk, and Zahl showed that a \\(K_{u,u}\\)-free semialgebraic graph on \\(N\\) vertices has at most \\(O_{d,D,u}(N^{2d/(d+1)+o(1)})\\) edges and Do extended this result to \\(K_{u,\\ldots,u}^{(k)}\\)-free semialgebraic hypergraphs. We improve upon both of these results by removing the \\(o(1)\\) in the exponent and making the dependence on \\(D\\) and \\(u\\) explicit and polynomial. All three of these results follow from a novel ``multilevel polynomial partitioning scheme'' that efficiently partitions a point set \\(P\\subset\\mathbb{R}^d\\) via low-complexity semialgebraic pieces. We prove this result using the polynomial method over varieties as developed by Walsh which extends the real polynomial partitioning technique of Guth and Katz. We give additional applications to the unit distance problem, the Erdős--Hajnal problem for semialgebraic graphs, and property testing of semialgebraic hypergraphs.

The Kruskal--Katona theorem determines the maximum number of \\(d\\)-cliques in an \\(n\\)-edge \\((d-1)\\)-uniform hypergraph. A generalization of the theorem was proposed by Bollobás and Eccles, called the partial shadow problem. The problem asks to determine the maximum number of \\(r\\)-sets of vertices that contain at least \\(d\\) edges in an \\(n\\)-edge \\((d-1)\\)-uniform hypergraph. In our previous work, we obtained an asymptotically tight upper bound via its connection to the joints problem, a problem in incidence geometry. In a different direction, Friedgut and Kahn generalized the Kruskal--Katona theorem by determining the maximum number of copies of any fixed hypergraph in an \\(n\\)-edge hypergraph, up to a multiplicative factor. In this paper, using the connection to the joints problem again, we generalize our previous work to show an analogous partial shadow phenomenon for any hypergraph, generalizing Friedgut and Kahn's result. The key idea is to encode the graph-theoretic problem with new kinds of joints that we call hypergraph joints. Our main theorem is a generalization of the joints theorem that upper bounds the number of hypergraph joints, which the partial shadow phenomenon immediately follows from. In addition, with an appropriate notion of multiplicities, our theorem also generalizes a generalization of H\"older's inequality considered by Finner.