Global homeomorphism theorem for manifolds and polyhedra

We improved a version of Global Homeomorphism Theorem due to Katriel such that it can be applied to more general geometric objects: Hilbert Riemannian manifolds, graphs, and polyhedra.Mathematics, AppliedMathematicsSCI(E)0ARTICLE97-10610

Identification of an Autoinhibitory Domain of p21-activated Protein Kinase 5

The p21-activated protein kinases (Paks) are serine/threonine protein kinases activated by binding to Rho family small GTPases, Rac and Cdc42. Recently, Pak family members have been subdivided into two groups, I and II. Group II Paks, including Pak4, Pak5, and Pak6, does not contain the highly conserved autoinhibitory domain that is found in the group I Paks members, i.e. Pak1, Pak2, and Pak3. In the present study, we have purified the glutathione S-transferase fusion form of Pak5 and shown for the first time that Pak5 autophosphorylation can be activated by GTP bound form of Cdc42. Mutation of histidine residues 19 and 22 to leucine on the p21-binding domain of Pak5 completely abolished the binding of Cdc42 and the Cdc42-mediated autophosphorylation. On the other hand, mutation of tyrosine 40 to cysteine of Cdc42 did not knockout the binding of Pak5. Analysis of C-terminal deletion mutants has identified an autoinhibitory fragment of Pak5 that is absent from other group II Pak family members. Taken together, these results suggest that Pak5, like Pak1, contains an autoinhibitory domain and its activity is regulated by Cdc42.postprin

Nonlinear Sampling Theory and Efficient Signal Recovery

Sampling theory investigates signal recovery from its partial information, and one of the simplest and most well-known sampling schemes is uniform linear sampling, characterized by the celebrated classical sampling theorem. However, the requirements of uniform linear sampling may not always be satisfied, sparking the need for more general sampling theories. In the thesis, we discuss the following three sampling scenarios: signal quantization, compressive sensing, and deep neural networks. In signal quantization theory, the inability of digital devices to perfectly store analog samples leads to distortion when reconstructing the signal from its samples. Different quantization schemes are proposed so as to minimize such distortion. We adapt a quantization scheme used in analog-to-digital conversion called signal decimation to finite dimensional signals. In doing so, we are able to achieve theoretically optimal reconstruction error decay rate. Compressive sensing investigates the possibility to recover high-dimensional signals from incomplete samples. It has been proven feasible as long as the signal is sufficiently sparse. To this point, all of the most successful examples follow from random constructions rather than deterministic ones. Whereas the sparsity of the signal can be almost as large as the ambient dimension for random constructions, current deterministic constructions require the sparsity to be at most the square-root of the ambient dimension. This apparent barrier is the well-known square-root bottleneck. In this thesis, we propose a new explicit sampling scheme as a possible candidate for deterministic compressive sensing. We present a partial result, while the full generality is still work in progress. For deep neural networks, one approximates signals with neural networks. To do so, many samples need to be drawn in order to find an optimal approximating neural network. A common approach is to employ stochastic gradient descent, but it is unclear if the resulting neural network is indeed optimal due to the non-convexity of the optimization scheme. We follow an alternative approach, utilizing the derivatives of the signal for stable reconstruction. In this thesis, we focus on non-smooth signals, and using weak differentiation, it is easy to obtain stable reconstruction for one-layer neural networks. We are currently working on the two-layer case, and our approach is outlined in this thesis

PHOTOLYTIC GENERATION OF NITRENIUM IONS: KINETIC STUDIES AND POLYMERIZATION REACTIONS

Nitrenium ions are highly reactive transient species that contain a positively charged and dicoordinate nitrogen atom. The nitrogen atom contains only six electrons in its valence shell and thus the nitrogen is electron deficient and bears a positive charge. Nitrenium ions are of interest due to their suspected role in carcinogenesis since amines are known to form covalent bonds to DNA. The synthesis and photolysis of 1-(N-methyl-N-(1-naphthyl)amino)-2,4,6-trimethylpyridinium tetrafluoroborate, by laser flash photolysis, allowed for the direct observation of N-methyl-1-naphthylnitrenium ion as well as measurements of N-methyl-1-naphthylnitrenium ion's lifetime and trapping rate constants. It was determined that N-methyl-1-naphthylnitrenium ion has an absorption maximum centered around 500 nm and a lifetime of 835 ns. The trapping rate constants with simple nucleophiles, such as chloride, alcohols, and amines, were determined to be on the order of 108 - 109 M-1s-1. These trapping rate constants were compared to other arylnitrenium ion systems to determine what factors contribute to a chemical's inherent carcinogenicity. The synthesis and photolysis of 1-(N-methyl-N-(2-naphthyl)amino)-2,4,6-trimethylpyridinium tetrafluoroborate was also performed. Although no transient intermediate was observed, products from photolysis are consistent with arylnitrenium ion products. Nitrenium ions are also of interest due to their possible role in the polymerization of aniline to form polyaniline (PANI). PANI is of interest because of it is an electronically conducting polymer with many commercial aspects and the mechanism of formation has been under dispute for decades. Although it is generally agreed that the initial dimerization step is due to radical cation coupling, the mechanism for aniline polymerization is argued as proceeding through either a radical cation mechanism or via a nitrenium ion mechanism. Synthesis of a photochemical precursors of an aniline dimer 4-(N-anilino)phenyl azide produced what is believed to be a 4-(N-anilino)phenylnitrenium ion which has an absorption maximum centered around 490 nm. Spectroscopic analysis by MALDI-TOF-MS and X-ray photoelectron spectroscopy (XPS), shows that PANI is a photoproduct. The extrapolated data and results from similar systems, supports the hypothesis that polymerization involves a nitrenium ion intermediate