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Let ‖ A ‖ max : = max i , j ⁡ | A i , j | denote the maximum magnitude of entries of a given matrix A. In this paper we show that max ⁡ { ‖ U r ‖ max , ‖ V r ‖ max } ≤ ( C r ) 6 r N , where U r and V r are the matrices whose columns are, respectively, the left and right singular vectors of the r-th order finite difference matrix D r with r ≥ 2 , and where D is the N × N finite difference matrix with 1 on the diagonal, −1 on the sub-diagonal, and 0 elsewhere. Here C is a universal constant that is independent of both N and r. Among other things, this establishes that both the right and left singular vectors of such finite difference matrices are Bounded Orthonormal Systems (BOSs) with known upper bounds on their BOS constants, objects of general interest in classical compressive sensing theory. Such finite difference matrices are also fundamental to standard r th order Sigma-Delta quantization schemes more specifically, and as a result the new bounds provided herein on the maximum l ∞ -norms of their l 2 -normalized singular vectors allow for several previous Sigma-Delta quantization results to be generalized and improved.

Using combinatorial methods, we determine that a projective coordinate ring of the moduli of parabolic principal $\mathrm{SL}_2$-bundles on a marked projective curve is not Gorenstein when the genus and number of marked points are greater than $1$., 12 pages, 6 figures, fixed an error in the main theorem, to appear in Electronic Journal of Combinatorics

2,3-Diarylfuro[2,3-b]pyridine-4-amines are a novel class of potent and selective inhibitors of Lck. The discovery, synthesis, and structure activity relationships of this series of inhibitors are reported. The most promising compounds were also profiled to deduce their pharmacokinetic properties.

Lck, or lymphocyte specific kinase, is a cytoplasmic tyrosine kinase of the Src family expressed in T-cells and NK cells. Genetic evidence from knockout mice and human mutations demonstrates that Lck kinase activity is critical for T-cell receptor-mediated signaling, leading to normal T-cell development and activation. A small molecule inhibitor of Lck is expected to be useful in the treatment of T-cell-mediated autoimmune and inflammatory disorders and/or organ transplant rejection. In this paper, we describe the structure-guided design, synthesis, structure-activity relationships, and pharmacological characterization of 2-amino-6-phenylpyrimido[5',4':5,6]pyrimido[1,2- a]benzimidazol-5(6 H)-ones, a new class of compounds that are potent inhibitors of Lck. The most promising compound of this series, 6-(2,6-dimethylphenyl)-2-((4-(4-methyl-1-piperazinyl)phenyl)amino)pyrimido[5',4':5,6]pyrimido-[1,2- a]benzimidazol-5(6 H)-one ( 25), exhibits potent inhibition of Lck kinase activity. This activity translates into inhibition of in vitro cell-based assays and in vivo models of T-cell activation and arthritis, respectively.

4-Amino-5,6-biaryl-furo[2,3-d]pyrimidines were identified as potent non-selective inhibitors of Lck. A novel, divergent, and practical synthetic route was developed to access derivatives from bifunctional intermediates. Lead optimization was guided by X-ray crystallographic data, and preliminary SAR led to the identification of compounds with improved cellular potency and selectivity.

The lymphocyte-specific kinase (Lck) is a cytoplasmic tyrosine kinase of the Src family expressed in T cells and NK cells. Genetic evidence in both mice and humans demonstrates that Lck kinase activity is critical for signaling mediated by the T cell receptor (TCR), which leads to normal T cell development and activation. A small molecule inhibitor of Lck is expected to be useful in the treatment of T cell-mediated autoimmune and inflammatory disorders and/or organ transplant rejection. In this paper, we describe the synthesis, structure-activity relationships, and pharmacological characterization of 2-aminopyrimidine carbamates, a new class of compounds with potent and selective inhibition of Lck. The most promising compound of this series, 2,6-dimethylphenyl 2-((3,5-bis(methyloxy)-4-((3-(4-methyl-1-piperazinyl)propyl)oxy)phenyl)amino)-4-pyrimidinyl(2,4-bis(methyloxy)phenyl)carbamate (43) exhibits good activity when evaluated in in vitro assays and in an in vivo model of T cell activation.