The value distribution of incomplete Gauss sums

It is well known that the classical Gauss sum, normalized by the square-root number of terms, takes only finitely many values. If one restricts the range of summation to a subinterval, a much richer structure emerges. We prove a limit law for the value distribution of such incomplete Gauss sums. The limit distribution is given by the distribution of a certain family of periodic functions. Our results complement Oskolkov's pointwise bounds for incomplete Gauss sums as well as the limit theorems for quadratic Weyl sums (theta sums) due to Jurkat and van Horne and the second author

On some families of invariant polynomials divisible by three and their zeta functions

In this note, we establish an analog of the Mallows-Sloane bound for Type III formal weight enumerators. This completes the bounds for all types (Types I through IV) in synthesis of our previous results. Next we show by using the binomial moments that there exists a family of polynomials divisible by three, which are not related to linear codes but are invariant under the MacWilliams transform for the value 3/2. We also discuss some properties of the zeta functions for such polynomials

Large inductive dimension of the Smirnov remainder

The purpose of this paper is to investigate the large inductive dimension of the remainder of the Smirnov compactification of the n-dimensional Euclidean space with the usual metric, and give an application of it.Comment: 8 pages, accepted for publication in Houston Journal of Mathematic

Progress in Selective Hydrogenation: Reactivity Trends in PdAu Heterogeneous Catalysts & Developing Chiral Building Blocks for New Homogeneous Catalysts

Selective hydrogenation is an essential method for converting cheap starting materials into more valuable chemical building blocks. Whether regioselectivity or diastereoselectivity is desired, catalysts are often employed to aid in this transformation. This work explores possibilities for finetuning both homogeneous and heterogeneous catalysts. Previously, bimetallic PdAu catalysts were used to enable the conversion of biologically produced 4-hydroxycoumarin (4HC) to pharmaceutically relevant coumarin and dihydroxycoumarin (DHC), but the subtleties of catalyst preparation and the effect of Pd loading were not known. Part I focuses on identifying reactivity trends resulting from variations in PdAu catalysts. Several PdAu catalysts with varying Pd:Au ratios were prepared by controlled surface reactions (CSR) or incipient wetness impregnation (IWI). Using high performance liquid chromatography (HPLC) and gas chromatography-mass spectrometry (GCMS), we determined the effects of the preparation method and Pd loading on reactivity, selectivity, and resistance to deactivation for the selective hydrogenation of 4HC to DHC. Part II aims to develop chiral building blocks for C1-symmetric, P-stereogenic ligands that will be used in the asymmetric hydrogenation of heavily functionalized substrates. To date, a new diastereoselective method of adding RO-H across a P=P double bond was identified and a new sterically dominating R group was generated. The results were analyzed by nuclear magnetic resonance (NMR) spectroscopy.Chemistr