Use of RP-TLC and Theoretical Computational Methods to Compare the Lipophilicity of Phenolic Drugs

JPC – Journal of Planar Chromatography – Modern TLC - The chromatographic behavior of the phenolic drugs niclosamide, hexachlorophene, ibuprofen, pentazocine, ethamivan, bithionol,...     

On the representation theory for full decomposable forms

Let $\text{k}\text{Q}$ be any finite normal extension and fix an order D of k invariant under the galois group $\text{G(}\text{k}\text{Q}\text{)}$. The ring class field K corresponding to D is normal over Q. We solve the problem of determining which full decomposable forms associated to the invertible ideals of D integrally represent a given positive integer m. First we establish a one-to-one correspondence between the improperly equivalent classes of such forms and the conjugacy classes of $\text{G(}\text{K}\text{Q}\text{)}$ which are contained in $\text{G(}\text{K}\text{k}\text{)}$. The solution is then seen to rest upon determining the Artin class in $\text{G(}\text{K}\text{Q}\text{)}$ of the unramified primes dividing m. This is accomplished by evaluating certain induced characters of $\text{G(}\text{K}\text{Q}\text{)}$ congruentially in terms of an associated integer linear recurrence sequence.     

On the minimal polynomial of Gauss periods for prime powers

For a positive integer , set and let denote the group of reduced residues modulo . Fix a congruence group of conductor and of order . Choose integers to represent the cosets of in . The Gauss periods

corresponding to are conjugate and distinct over with minimal polynomial

To determine the coefficients of the period polynomial (or equivalently, its reciprocal polynomial is a classical problem dating back to Gauss. Previous work of the author, and Gupta and Zagier, primarily treated the case , an odd prime, with fixed. In this setting, it is known for certain integral power series and , that for any positive integer

holds in for all primes except those in an effectively determinable finite set. Here we describe an analogous result for the case , a prime power ( ). The methods extend for odd prime powers to give a similar result for certain twisted Gauss periods of the form

where denotes the usual Legendre symbol and .

     

Controlling cyclization pathways in palladium(ii)-catalyzed intramolecular alkene hydro-functionalization via substrate directivity

We report a series of palladium(ii)-catalyzed, intramolecular alkene hydrofunctionalization reactions with carbon, nitrogen, and oxygen nucleophiles to form five- and six-membered carbo- and heterocycles. In these reactions, the presence of a proximal bidentate directing group controls the cyclization pathway, dictating the ring size that is generated, even in cases that are disfavored based on Baldwin''s rules and in cases where there is an inherent preference for an alternative pathway. DFT studies shed light on the origins of pathway selectivity in these processes.

We report a series of palladium(ii)-catalyzed, intramolecular alkene hydrofunctionalization reactions with carbon, nitrogen, and oxygen nucleophiles to form five- and six-membered carbo- and heterocycles.     

Explicit values of multi-dimensional Kloosterman sums for prime powers, II

For any integer fix , and let denote the group of reduced residues modulo . Let , a power of a prime . The hyper-Kloosterman sums of dimension are defined for by

where denotes the multiplicative inverse of modulo .

Salie evaluated in the classical setting for even , and for odd with . Later, Smith provided formulas that simplified the computation of in these cases for . Recently, Cochrane, Liu and Zheng computed upper bounds for in the general case , stopping short of their explicit evaluation. Here I complete the computation they initiated to obtain explicit values for the Kloosterman sums for , relying on basic properties of some simple specialized exponential sums. The treatment here is more elementary than the author's previous determination of these Kloosterman sums using character theory and -adic methods. At the least, it provides an alternative, independent evaluation of the Kloosterman sums.

     

On the rational equivalence of full decomposable forms

Let k be any finite normal extension of the rational field Q and fix an order D of k invariant under the galois group $\text{G(}\text{k}\text{Q}\text{)}$. Consider the set F of the full decomposable forms which correspond to the invertible fractional ideals of D. In a recent paper the author has given arithmetic criteria to determine which classes of improperly equivalent forms in F integrally represent a given positive rational integer m. These criteria are formulated in terms of certain integer sequences which satisfy a linear recursion and need only be considered modulo the primes dividing m. Here, for the most part, we consider partitioning F under rational equivalence. It is a found that the set of rationally equivalent classes in F is a group under composition of forms analogous to Gauss' and Dirichlet's classical results for binary quadratic forms. This leads us to given criteria as before to determine which classes of rationally equivalent forms in F rationally represent m. Moreover, by applying the genus theory of number fields, we find arithmetic criteria to determine when everywhere local norms are global norms if the Hasse norm principle fails to hold in $\text{k}\text{Q}$.