A multiple-instance scoring method to predict tissue-specific cis-regulatory motifs and regions

Transcription is the central process of gene regulation. In higher eukaryotes, the transcription of a gene is usually regulated by multiple cis-regulatory regions (CRRs). In different tissues, different transcription factors bind to their cis-regulatory motifs in these CRRs to drive tissue-specific expression patterns of their target genes. By combining the genome-wide gene expression data with the genomic sequence data, we proposed multiple-instance scoring (MIS) method to predict the tissue-specific motifs and the corresponding CRRs. The method is mainly based on the assumption that only a subset of CRRs of the expressed gene should function in the studied tissue. By testing on the simulated datasets and the fly muscle dataset, MIS can identify true motifs when noise is high and shows higher specificity for predicting the tissue-specific functions of CRRs

Zero-error communication over adder MAC

Adder MAC is a simple noiseless multiple-access channel (MAC), where if users send messages $X_1,\ldots,X_h\in \{0,1\}^n$, then the receiver receives $Y = X_1+\cdots+X_h$ with addition over $\mathbb{Z}$. Communication over the noiseless adder MAC has been studied for more than fifty years. There are two models of particular interest: uniquely decodable code tuples, and $B_h$-codes. In spite of the similarities between these two models, lower bounds and upper bounds of the optimal sum rate of uniquely decodable code tuple asymptotically match as number of users goes to infinity, while there is a gap of factor two between lower bounds and upper bounds of the optimal rate of $B_h$-codes. The best currently known $B_h$-codes for $h\ge 3$ are constructed using random coding. In this work, we study variants of the random coding method and related problems, in hope of achieving $B_h$-codes with better rate. Our contribution include the following. (1) We prove that changing the underlying distribution used in random coding cannot improve the rate. (2) We determine the rate of a list-decoding version of $B_h$-codes achieved by the random coding method. (3) We study several related problems about R\'{e}nyi entropy.Comment: An updated version of author's master thesi

Spanning rigid subgraph packing and sparse subgraph covering

Rigidity, arising in discrete geometry, is the property of a structure that does not flex. Laman provides a combinatorial characterization of rigid graphs in the Euclidean plane, and thus rigid graphs in the Euclidean plane have applications in graph theory. We discover a sufficient partition condition of packing spanning rigid subgraphs and spanning trees. As a corollary, we show that a simple graph $G$ contains a packing of $k$ spanning rigid subgraphs and $l$ spanning trees if $G$ is $(4k+2l)$-edge-connected, and $G-Z$ is essentially $(6k+2l - 2k|Z|)$-edge-connected for every $Z\subset V(G)$. Thus every $(4k+2l)$-connected and essentially $(6k+2l)$-connected graph $G$ contains a packing of $k$ spanning rigid subgraphs and $l$ spanning trees. Utilizing this, we show that every $6$-connected and essentially $8$-connected graph $G$ contains a spanning tree $T$ such that $G-E(T)$ is $2$-connected. These improve some previous results. Sparse subgraph covering problems are also studied.Comment: 12 page

Computations of Mather Minimal Log Discrepancies

We compute the Mather minimal log discrepancy via jet schemes and arc spaces for toric varieties and very general hypersurfaces