Graph Embeddings via Tensor Products and Approximately Orthonormal Codes

We introduce a method for embedding graphs as vectors in a structure-preserving manner, showcasing its rich representational capacity and giving some theoretical properties. Our procedure falls under the bind-and-sum approach, and we show that our binding operation - the tensor product - is the most general binding operation that respects the principle of superposition. We also establish some precise results characterizing the behavior of our method, and we show that our use of spherical codes achieves a packing upper bound. Then, we perform experiments showcasing our method's accuracy in various graph operations even when the number of edges is quite large. Finally, we establish a link to adjacency matrices, showing that our method is, in some sense, a generalization of adjacency matrices with applications towards large sparse graphs.Comment: 20 pages, 2 figure

Commutativity and Disentanglement from the Manifold Perspective

In this paper, we interpret disentanglement as the discovery of local charts and trace how that definition naturally leads to an equivalent condition for disentanglement: the disentangled factors must commute with each other. We discuss the practical and theoretical implications of commutativity, in particular the compression and disentanglement of generative models. Finally, we conclude with a discussion of related approaches to disentanglement and how they relate to our view of disentanglement from the manifold perspective.Comment: 20 pages, 1 tabl

Random Projections of Sparse Adjacency Matrices

We analyze a random projection method for adjacency matrices, studying its utility in representing sparse graphs. We show that these random projections retain the functionality of their underlying adjacency matrices while having extra properties that make them attractive as dynamic graph representations. In particular, they can represent graphs of different sizes and vertex sets in the same space, allowing for the aggregation and manipulation of graphs in a unified manner. We also provide results on how the size of the projections need to scale in order to preserve accurate graph operations, showing that the size of the projections can scale linearly with the number of vertices while accurately retaining first-order graph information. We conclude by characterizing our random projection as a distance-preserving map of adjacency matrices analogous to the usual Johnson-Lindenstrauss map.Comment: 21 page

Dynamic changes in connexin expression correlate with key events in the wound healing process.

Wound healing is a complex process requiring communication for the precise co-ordination of different cell types. The role of extracellular communication through growth factors in the wound healing process has been extensively documented, but the role of direct intercellular communication via gap junctions has scarcely been investigated. We have examined the dynamics of gap junction protein (Connexins 26, 30, 31.1 and 43) expression in the murine epidermis and dermis during wound healing, and we show that connexin expression is extremely plastic between 6 hours and 12 days post-wounding. The immediate response (6 h) to wounding is to downregulate all connexins in the epidermis, but thereafter the expression profile of each connexin changes dramatically. Here, we correlate the changing patterns of connexin expression with key events in the wound healing process

Inseparable gersgorin discs and the existence of conjugate complex eigenvalues of real matrices

We investigate the converse of the known fact that if the Gersgorin discs of a real n-by-n matrix may be separated by positive diagonal similarity then the eigenvalues are real. In the 2-by-2 case, with appropriate signs for the off-diagonal entries, we find that the converse is correct, which raises several questions. First, in the 3-by-3 case, the converse is not generally correct, but, empirically it is frequently true. Then, in the n-by-n case, n >=3, we find that if all the 2-by-2 principal submatrices have inseparable discs (\strongly inseparable discs"), the full matrix must have a nontrivial pair of conjugate complex eigenvalues (i.e., cannot have all real eigenvalues). This hypothesis cannot generally be weakened.NSF grant DMS #075196

Advance the DNA computing

It has been previously shown that DNA computing can solve those problems currently intractable on even the fastest electronic computers. The algorithm design for DNA computing, however, is not straightforward. A strong background in both the DNA molecule and computer engineering are required to develop efficient DNA computing algorithms. After Adleman solved the Hamilton Path Problem using a combinatorial molecular method, many other hard computational problems were investigated with the proposed DNA computer. The existing models from which a few DNA computing algorithms have been developed are not sufficiently powerful and robust, however, to attract potential users. This thesis has described research performed to build a new DNA computing model based on various new algorithms developed to solve the 3-Coloring problem. These new algorithms are presented as vehicles for demonstrating the advantages of the new model, and they can be expanded to solve other NP-complete problems. These new algorithms can significantly speed up computation and therefore achieve a consistently better time performance. With the given resource, these algorithms can also solve problems of a much greater size, especially as compared to existing DNA computation algorithms. The error rate can also be greatly reduced by applying these new algorithms. Furthermore, they have the advantage of dynamic updating, so an answer can be changed based on modifications made to the initial condition. This new model makes use of the huge possible memory by generating a ``lookup table'' during the implementation of the algorithms. If the initial condition changes, the answer changes accordingly. In addition, the new model has the advantage of decoding all the strands in the final pool both quickly and efficiently. The advantages provided by the new model make DNA computing an efficient and attractive means of solving computationally intense problems

Tris(4-azidophenyl)methanol - a novel and multifunctional thiol protecting group

The novel tris(4-azidophenyl)methanol, a multifunctionalisable aryl azide, is reported. The aryl azide can be used as a protecting group for thiols in peptoid synthesis and can be cleaved under mild reaction conditions via a Staudinger reduction. Moreover, the easily accessible aryl azide can be functionalised via copper-catalysed cycloaddition reactions, providing additional opportunities for materials chemistry applications.Peer reviewe

Internal structure of the San Jacinto fault zone at Blackburn Saddle from seismic data of a linear array

Local and teleseismic earthquake waveforms recorded by a 180-m-long linear array (BB) with seven seismometers crossing the Clark fault of the San Jacinto fault zone northwest of Anza are used to image a deep bimaterial interface and core damage structure of the fault. Delay times of P waves across the array indicate an increase in slowness from the southwest most (BB01) to the northeast most (BB07) station. Automatic algorithms combined with visual inspection and additional analyses are used to identify local events generating fault zone head and trapped waves. The observed fault zone head waves imply that the Clark fault in the area is a sharp bimaterial interface, with lower seismic velocity on the southwest side. The moveout between the head and direct P arrivals for events within ∼40 km epicentral distance indicates an average velocity contrast across the fault over that section and the top 20 km of 3.2 per cent. A constant moveout for events beyond ∼40 km to the southeast is due to off-fault locations of these events or because the imaged deep bimaterial interface is discontinuous or ends at that distance. The lack of head waves from events beyond ∼20 km to the northwest is associated with structural complexity near the Hemet stepover. Events located in a broad region generate fault zone trapped waves at stations BB04–BB07. Waveform inversions indicate that the most likely parameters of the trapping structure are width of ∼200 m, S velocity reduction of 30–40 per cent with respect to the bounding blocks, Q value of 10–20 and depth of ∼3.5 km. The trapping structure and zone with largest slowness are on the northeast side of the fault. The observed sense of velocity contrast and asymmetric damage across the fault suggest preferred rupture direction of earthquakes to the northwest. This inference is consistent with results of other geological and seismological studies