Constructing nonresidues in finite fields and the extended Riemann hypothesis

We present a new deterministic algorithm for the problem of constructing th power nonresidues in finite fields , where is prime and is a prime divisor of . We prove under the assumption of the Extended Riemann Hypothesis (ERH), that for fixed and , our algorithm runs in polynomial time. Unlike other deterministic algorithms for this problem, this polynomial-time bound holds even if is exponentially large. More generally, assuming the ERH, in time we can construct a set of elements that generates the multiplicative group . An extended abstract of this paper appeared in Proc. 23rd Ann. ACM Symp. on Theory of Computing, 1991.

     

2-Designs and a differential equation

We show that 2-designs with given parameters v, k, are in one-to-one correspondence to polynomials that solve a certain differential equation and have coefficients equal to zero or one. From this result we derive an existence theorem whereby designs correspond to integer points on a sphere in Euclidean space.     

Notiz zur Darstellung von 4-Hydroxy-6-methyl-5,6-dihydro-2H-pyran-2-on

Meldrum's acid (2) is acylated by diketen affording the acylMeldrum's acid3. In aqueous bicarbonate its 3-oxogroup is selectively reduced by sodium tetrahydroborate giving the alcohol4, which readily undergoes cyclization in refluxing dioxane leading to the title pyrone7. Under identical conditions3 produces the pyrone6 with the pyrone carboxylic acid5 as intermediate.

     

Resonances and microlocal analysis

In this paper we outline briefly how microlocal analysis can be applied to give a general approach to the mathematical theory of resonances in the semiclassical limit. We also describe recent results about the asymptotic behavior of resonances generated by closed trajectories and stationary points in the classical flow.     

The Donor-Acceptor-Acceptor Purine Analog: Transformation of 5-aza-7-deaza-1H-isoguanine (=4-aminoimidazo-[1,2-a]-1,3,5-triazin-2(1H)-one) to 2′-deoxy-5-aza-7-deaza-isoguanosine using purine nucleoside phosphorylase

A new synthesis is reported for 4-aminoimidazo[1,2-a]-1,3,5-triazin-2(1H)-one ( =5-aza-7-deaza-isoguanosine; 8 ), a purine analog that, when incorporated into an oligonucleotide chain, presents a H-bond donor-acceptor-acceptor pattern to a complementary pyrimidine analog. A protected ribose derivative was coupled to 8 to yield 4-amino-8-(β-D -ribofuranosyl)imidazo[1,2-a]-1,3,5-triazin-2(8H)-one ( =5-aza-7-deaza-isoguanosine; 11 ) after deprotection, Alternatively, direct synthesis of both the ribo derivative 11 and the corresponding deoxyribo derivative 17 as the β-D -anomers was achieved using the enzyme purine nucleoside phosphorylase in a one-pot reaction. This adapts a known synthetic approach to yield a new strategy for obtaining diastereoisomerically pure deoxyribonucleoside analogs on 1-gram scales.